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Riemann hypothesis

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Conjecture on zeros of the zeta function

For the musical term, seeRiemannian theory.

Unsolved problem in mathematics

Do all non-trivial zeros of the Riemann zeta function have a real part equal to one half?

Millennium Prize Problems
Birch and Swinnerton-Dyer conjecture Hodge conjecture Navier–Stokes existence and smoothness P versus NP problem Poincaré conjecture (solved) Riemann hypothesis Yang–Mills existence and mass gap
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Riemann zeta function

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TheRiemann zeta function is defined for complex $s {\displaystyle s}$ with real part greater than 1 by theabsolutely convergent infinite series

\zeta (s)=\sum _{n=1}^{\infty }{\frac {1}{n^{s}}}={\frac {1}{1^{s}}}+{\frac {1}{2^{s}}}+{\frac {1}{3^{s}}}+\cdots

Leonhard Euler considered this series in the 1730s for real values of $s {\displaystyle s}$ , in conjunction with his solution to theBasel problem. He also proved that it equals theEuler product

\zeta (s)=\prod _{p{\text{ prime}}}{\frac {1}{1-p^{-s}}}={\frac {1}{1-2^{-s}}}\cdot {\frac {1}{1-3^{-s}}}\cdot {\frac {1}{1-5^{-s}}}\cdot {\frac {1}{1-7^{-s}}}\cdots

where theinfinite product extends over all prime numbers $p {\displaystyle p}$ .^[2]

The Riemann hypothesis discusses zeros outside theregion of convergence of this series and Euler product. To make sense of the hypothesis, it is necessary toanalytically continue the function to obtain a form that is valid for all complex $s {\displaystyle s}$ . Because the zeta function ismeromorphic, all choices of how to perform this analytic continuation will lead to the same result, by theidentity theorem. A first step in this continuation observes that the series for the zeta function and theDirichlet eta function satisfy the relation

\left(1-{\frac {2}{2^{s}}}\right)\zeta (s)=\eta (s)=\sum _{n=1}^{\infty }{\frac {(-1)^{n+1}}{n^{s}}}={\frac {1}{1^{s}}}-{\frac {1}{2^{s}}}+{\frac {1}{3^{s}}}-\cdots ,

within the region of convergence for both series. But the eta function series on the right converges not just when the real part of $s {\displaystyle s}$ is greater than one, but more generally whenever $s {\displaystyle s}$ has positive real part. Thus, the zeta function can be redefined as $\eta (s)/(1-2/2^{s})$ , extending it from $\operatorname {Re} (s)>1$ to the larger domain $\operatorname {Re} (s)>0$ , except for the points where $1-2/2^{s}$ is zero. These are the points $s=1+2\pi in/\log 2$ , where $n {\displaystyle n}$ can be any nonzero integer; the zeta function can be extended to these values too by taking limits (see the article on theDirichlet eta function), giving a finite value for all values of $s {\displaystyle s}$ with positive real part except thesimple pole at $s=1$ .

In the strip $0<\operatorname {Re} (s)<1$ this extension of the zeta function satisfies thefunctional equation

\zeta (s)=2^{s}\pi ^{s-1}\ \sin \left({\frac {\pi s}{2}}\right)\ \Gamma (1-s)\ \zeta (1-s).

One may then define $\zeta (s)$ for all remaining nonzero complex numbers $s {\displaystyle s}$ ( $\operatorname {Re} (s)\leq 0$ and $s\neq 0$ ) by applying this equation outside the strip, and letting $\zeta (s)$ equal the right side of the equation whenever $s {\displaystyle s}$ has non-positive real part (and $s\neq 0$ ).

If $s {\displaystyle s}$ is a negative even integer, then $\zeta (s)=0$ , because the factor $\sin(\pi s/2)$ vanishes; these are the zeta function'strivial zeros. (If $s {\displaystyle s}$ is a positive even integer this argument does not apply because the zeros of thesine function are canceled by the poles of thegamma function as it takes negative integer arguments.)

The valueζ(0) = −1/2 is not determined by the functional equation, but is the limiting value of $\zeta (s)$ as $s {\displaystyle s}$ approaches zero. The functional equation also implies that the zeta function has no zeros with negative real part other than the trivial zeros, so all nontrivial zeros lie in thecritical strip where $s {\displaystyle s}$ has real part between 0 and 1.

Riemann zeta function along the critical line withRe(s) = 1/2. Real values are shown on the horizontal axis and imaginary values are on the vertical axis.Re(ζ(1/2 +it)),Im(ζ(1/2 +it)) is plotted witht ranging between −30 and 30.^[3]
3D animation showing critical strip (blue, where $s {\displaystyle s}$ has real part between 0 and 1), critical line (red, for real part of $s {\displaystyle s}$ equals 0.5) and zeroes (cross between red and orange): [x,y,z] = [Re(ζ(r +it)), Im(ζ(r +it)),t] with $0.1\leq r\leq 0.9$ and $1\leq t\leq 51$ .
The real part (red) and imaginary part (blue) of the Riemann zeta function $\zeta (s)$ along the critical line in thecomplex plane with real part $\operatorname {Re} (s)=1/2$ . The firstnontrivial zeros, where $\zeta (s)$ equals zero, occur where both curves touch the horizontal $x {\displaystyle x}$ -axis, for complex numbers with imaginary parts equaling $\pm 14.135$ , $\pm 21.022$ and $\pm 25.011$ .

Origin

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... es ist sehr wahrscheinlich, dass alle Wurzeln reell sind. Hiervon wäre allerdings ein strenger Beweis zu wünschen; ich habe indess die Aufsuchung desselben nach einigen flüchtigen vergeblichen Versuchen vorläufig bei Seite gelassen, da er für den nächsten Zweck meiner Untersuchung entbehrlich schien.

... it is very probable that all roots are real. Of course one would wish for a rigorous proof here; I have for the time being, after some fleeting vain attempts, provisionally put aside the search for this, as it appears dispensable for the immediate objective of my investigation.

— Riemann's statement of the Riemann hypothesis, from (Riemann 1859). (He was discussing a variant of the zeta function, modified in a way that the real line be mapped to the critical line.)

At the death of Riemann, a note was found among his papers, saying "These properties ofζ(s) (the function in question) are deduced from an expression of it which, however, I did not succeed in simplifying enough to publish it."We still have not the slightest idea of what the expression could be. As to the properties he simply enunciated, some thirty years elapsed before I was able to prove all of them but one [the Riemann Hypothesis itself].

— Jacques Hadamard, The Mathematician's Mind, VIII. Paradoxical Cases of Intuition

Riemann's original motivation for studying the zeta function and its zeros was their occurrence in hisexplicit formula for thenumber of primes $\pi (x)$ less than or equal to a given number $x {\displaystyle x}$ , which he published in his 1859 paper "On the Number of Primes Less Than a Given Magnitude". His formula was given in terms of the related function

\Pi (x)=\pi (x)+{\frac {\pi (x^{1/2})}{2}}+{\frac {\pi (x^{1/3})}{3}}+{\frac {\pi (x^{1/4})}{4}}+{\frac {\pi (x^{1/5})}{5}}+{\frac {\pi (x^{1/6})}{6}}+\cdots

which counts the primes and prime powers up to $x {\displaystyle x}$ , counting a prime power $p^{n}$ as $1/n$ . The number of primes can be recovered from this function by using theMöbius inversion formula:

{\begin{aligned}\pi (x)&=\sum _{n=1}^{\infty }{\frac {\mu (n)}{n}}\Pi (x^{1/n})\\&=\Pi (x)-{\frac {1}{2}}\Pi (x^{1/2})-{\frac {1}{3}}\Pi (x^{1/3})-{\frac {1}{5}}\Pi (x^{1/5})+{\frac {1}{6}}\Pi (x^{1/6})-\cdots ,\end{aligned}}

where $\mu$ is theMöbius function. Riemann's formula is then

\Pi _{0}(x)=\operatorname {li} (x)-\sum _{\rho }\operatorname {li} (x^{\rho })-\log 2+\int _{x}^{\infty }{\frac {dt}{t(t^{2}-1)\log t}}

where the sum is over the nontrivial zeros of the zeta function and where $\Pi _{0}$ is a slightly modified version of $\Pi$ that replaces its value at its points ofdiscontinuity by the average of its upper and lower limits:

\Pi _{0}(x)=\lim _{\varepsilon \to 0}{\frac {\Pi (x-\varepsilon )+\Pi (x+\varepsilon )}{2}}.

The summation in Riemann's formula is not absolutely convergent, but may be evaluated by taking the zeros $\rho$ in order of the absolute value of their imaginary part. The function $\operatorname {li}$ occurring in the first term is the (unoffset)logarithmic integral function given by theCauchy principal value of the divergent integral

\operatorname {li} (x)=\int _{0}^{x}{\frac {dt}{\log t}}.

The terms $\operatorname {li} (x^{\rho })$ involving the zeros of the zeta function need some care in their definition as $\operatorname {li}$ has branch points at 0 and 1, and are defined (for $x>1$ ) by analytic continuation in the complex variable $\rho$ in the region $\operatorname {Re} (\rho )>0$ ; i.e., they should be considered asEi(ρ logx). The other terms also correspond to zeros: the dominant term $\operatorname {li} (x)$ comes from the pole at $s=1$ , considered as a zero of multiplicity $-1$ , and the remaining small terms come from the trivial zeros. For some graphs of the sums of the first few terms of this series seeRiesel & Göhl (1970) orZagier (1977).

This formula says that the zeros of the Riemann zeta function control theoscillations of primes around their "expected" positions. Riemann knew that the non-trivial zeros of the zeta function were symmetrically distributed about the line $s=1/2+it$ , and he knew that all of its non-trivial zeros must lie in the range $0\leq \operatorname {Re} (s)\leq 1$ . He checked that a few of the zeros lay on the critical line with real part $1/2$ and suggested that they all do; this is the Riemann hypothesis.

The result has caught the imagination of most mathematicians because it is so unexpected, connecting two seemingly unrelated areas in mathematics; namely,number theory, which is the study of the discrete, andcomplex analysis, which deals with continuous processes.

— (Burton 2006, p. 376)

Consequences

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The practical uses of the Riemann hypothesis include many propositions known to be true under the Riemann hypothesis, and some that can be shown to be equivalent to the Riemann hypothesis.

Distribution of prime numbers

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Riemann's explicit formula forthe number of primes less than a given number states that, in terms of a sum over the zeros of the Riemann zeta function, the magnitude of the oscillations of primes around their expected position is controlled by the real parts of the zeros of the zeta function. In particular, the error term in theprime number theorem is closely related to the position of the zeros. For example, if $\beta$ is theupper bound of the real parts of the zeros, then^[4] $\pi (x)-\operatorname {li} (x)=O\!\left(x^{\beta }\log x\right)$ , where $\pi (x)$ is theprime-counting function and $\operatorname {li} (x)$ is thelogarithmic integral function.It is already known that $1/2\leq \beta \leq 1$ .^[5]

Corrections to anestimate of the prime-counting function using zeros of the zeta function. The magnitude of the correction term is determined by the real part of the zero being added in the correction.

Helge von Koch proved that the Riemann hypothesis implies the "best possible" bound for the error of the prime number theorem.^[6] A precise version of von Koch's result, due toSchoenfeld (1976), says that the Riemann hypothesis implies

|\pi (x)-\operatorname {li} (x)|<{\frac {1}{8\pi }}{\sqrt {x}}\log(x)

for all $x\geq 2657$ .Schoenfeld (1976) also showed that the Riemann hypothesis implies

|\psi (x)-x|<{\frac {1}{8\pi }}{\sqrt {x}}\log ^{2}x

for all $x\geq 73.2$ , where $\psi (x)$ isChebyshev's second function.

Adrian Dudek^[7] proved that the Riemann hypothesis implies that for $x\geq 2$ , there is a prime $p {\displaystyle p}$ satisfying

x-{\frac {4}{\pi }}{\sqrt {x}}\log x<p\leq x

The constant $4/\pi$ may be reduced to $1+\varepsilon$ provided that $x {\displaystyle x}$ is taken to be sufficiently large. This is an explicit version of a theorem ofCramér.

Growth of arithmetic functions

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The Riemann hypothesis implies strong bounds on the growth of many otherarithmetic functions, in addition to the primes counting function above.

One example involves theMöbius functionμ. The statement that the equation

{\frac {1}{\zeta (s)}}=\sum _{n=1}^{\infty }{\frac {\mu (n)}{n^{s}}}

is valid for everys with real part greater than 1/2, with the sum on the right hand side converging, is equivalent to the Riemann hypothesis. From this we can also conclude that if theMertens function is defined by

M(x)=\sum _{n\leq x}\mu (n)

then the claim that

M(x)=O\left(x^{{\frac {1}{2}}+\varepsilon }\right)

for every positiveε is equivalent to the Riemann hypothesis (J. E. Littlewood, 1912; see for instance: paragraph 14.25 inTitchmarsh (1986)). Thedeterminant of the ordernRedheffer matrix is equal toM(n), so the Riemann hypothesis can also be stated as a condition on the growth of these determinants. Littlewood's result has been improved several times since then, byEdmund Landau,^[8]Edward Charles Titchmarsh,^[9] Helmut Maier andHugh Montgomery,^[10] andKannan Soundararajan.^[11] Soundararajan's result is that, conditional on the Riemann hypothesis,

M(x)=O\left(x^{1/2}\exp \left((\log x)^{1/2}(\log \log x)^{14}\right)\right).

The Riemann hypothesis puts a rather tight bound on the growth ofM, sinceOdlyzko & te Riele (1985) disproved the slightly strongerMertens conjecture

|M(x)|\leq {\sqrt {x}}.

Another closely related result is due toBjörner (2011), that the Riemann hypothesis is equivalent to the statement that theEuler characteristic of thesimplicial complex determined by the lattice of integers under divisibility is $o(n^{1/2+\epsilon })$ for all $\epsilon >0$ (seeincidence algebra).

The Riemann hypothesis is equivalent to many other conjectures about the rate of growth of other arithmetic functions aside fromμ(n). A typical example isRobin's theorem,^[12] which states that ifσ(n) is thesigma function, given by

\sigma (n)=\sum _{d\mid n}d

then

\sigma (n)<e^{\gamma }n\log \log n

for alln > 5040 if and only if the Riemann hypothesis is true, whereγ is theEuler–Mascheroni constant.

A related bound was given byJeffrey Lagarias in 2002, who proved that the Riemann hypothesis is equivalent to the statement that:

\sigma (n)<H_{n}+\log(H_{n})e^{H_{n}}

for everynatural numbern > 1, where $H_{n}$ is thenthharmonic number.^[13]

The Riemann hypothesis is also true if and only if the inequality

{\frac {n}{\varphi (n)}}<e^{\gamma }\log \log n+{\frac {e^{\gamma }(4+\gamma -\log 4\pi )}{\sqrt {\log n}}}

is true for alln ≥ 120569#, whereφ(n) isEuler's totient function and 120569# is theproduct of the first 120569 primes.^[14]

Another example was found byJérôme Franel, and extended byLandau (seeFranel & Landau (1924)). The Riemann hypothesis is equivalent to several statements showing that the terms of theFarey sequence are fairly regular. One such equivalence is as follows: ifF_n is the Farey sequence of ordern, beginning with 1/n and up to 1/1, then the claim that for allε > 0

\sum _{i=1}^{m}|F_{n}(i)-{\tfrac {i}{m}}|=O\left(n^{{\frac {1}{2}}+\epsilon }\right)

is equivalent to the Riemann hypothesis. Here

m=\sum _{i=1}^{n}\varphi (i)

is the number of terms in the Farey sequence of ordern.

For an example fromgroup theory, ifg(n) isLandau's function given by the maximal order of elements of thesymmetric group S_n of degreen, thenMassias, Nicolas & Robin (1988) showed that the Riemann hypothesis is equivalent to the bound

\log g(n)<{\sqrt {\operatorname {Li} ^{-1}(n)}}

for all sufficiently largen.

Lindelöf hypothesis and growth of the zeta function

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The Riemann hypothesis has various weaker consequences as well; one is theLindelöf hypothesis on the rate of growth of the zeta function on the critical line, which says that, for anyε > 0,

\zeta \left({\frac {1}{2}}+it\right)=O(t^{\varepsilon }),

ast → $\infty$ .

The Riemann hypothesis also implies quite sharp bounds for the growth rate of the zeta function in other regions of the critical strip. For example, it implies that

e^{\gamma }\leq \limsup _{t\rightarrow +\infty }{\frac {|\zeta (1+it)|}{\log \log t}}\leq 2e^{\gamma }

{\frac {6}{\pi ^{2}}}e^{\gamma }\leq \limsup _{t\rightarrow +\infty }{\frac {1/|\zeta (1+it)|}{\log \log t}}\leq {\frac {12}{\pi ^{2}}}e^{\gamma }

so the growth rate ofζ(1 +it) and its inverse would be known up to a factor of 2.^[15]

Large prime gap conjecture

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The prime number theorem implies that on average, thegap between the primep and its successor islogp. However, some gaps between primes may be much larger than the average. Cramér proved that, assuming the Riemann hypothesis, every gap isO(√p logp). This is a case in which even the best bound that can be proved using the Riemann hypothesis is far weaker than what seems true:Cramér's conjecture implies that every gap isO((logp)²), which, while larger than the average gap, is far smaller than the bound implied by the Riemann hypothesis. Numerical evidence supports Cramér's conjecture.^[16]

Analytic criteria equivalent to the Riemann hypothesis

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Many statements equivalent to the Riemann hypothesis have been found, though so far none of them have led to much progress in proving (or disproving) it. Some typical examples are as follows. (Others involve thedivisor functionσ(n).)

TheRiesz criterion was given byRiesz (1916), to the effect that the bound

-\sum _{k=1}^{\infty }{\frac {(-x)^{k}}{(k-1)!\zeta (2k)}}=O\left(x^{{\frac {1}{4}}+\epsilon }\right)

holds for all ε > 0 if and only if the Riemann hypothesis holds. See also theHardy–Littlewood criterion.

Nyman (1950) proved that the Riemann hypothesis is true if and only if the space of functions of the form

f(x)=\sum _{\nu =1}^{n}c_{\nu }\rho \left({\frac {\theta _{\nu }}{x}}\right)

whereρ(z) is the fractional part ofz,0 ≤θ_ν ≤ 1, and

\sum _{\nu =1}^{n}c_{\nu }\theta _{\nu }=0,

is dense in theHilbert spaceL²(0,1) of square-integrable functions on the unit interval.Beurling (1955) extended this by showing that the zeta function has no zeros with real part greater than 1/p if and only if this function space is dense inL^p(0,1). This Nyman-Beurling criterion was strengthened by Baez-Duarte^[17] to the case where $\theta _{\nu }\in \{1/k\}_{k\geq 1}$ .

Salem (1953) showed that the Riemann hypothesis is true if and only if the integral equation

\int _{0}^{\infty }{\frac {z^{-\sigma -1}\varphi (z)}{{e^{x/z}}+1}}\,dz=0

has no non-trivial bounded solutions $\varphi$ for $1/2<\sigma <1$ .

Weil's criterion is the statement that the positivity of a certain function is equivalent to the Riemann hypothesis. Related isLi's criterion, a statement that the positivity of a certain sequence of numbers is equivalent to the Riemann hypothesis.

Speiser (1934) proved that the Riemann hypothesis is equivalent to the statement thatζ′(s), the derivative ofζ(s), has no zeros in the strip

0<\Re (s)<{\frac {1}{2}}.

Thatζ(s) has only simple zeros on the critical line is equivalent to its derivative having no zeros on the critical line.

TheFarey sequence provides two equivalences, due toJerome Franel andEdmund Landau in 1924.

Thede Bruijn–Newman constant denoted by Λ and named afterNicolaas Govert de Bruijn andCharles M. Newman, is definedas the unique real number such that thefunction

H(\lambda ,z):=\int _{0}^{\infty }e^{\lambda u^{2}}\Phi (u)\cos(zu)\,du

that is parametrised by a real parameterλ, has a complex variablez and is defined using a super-exponentially decaying function

\Phi (u)=\sum _{n=1}^{\infty }(2\pi ^{2}n^{4}e^{9u}-3\pi n^{2}e^{5u})e^{-\pi n^{2}e^{4u}}

has only real zeros if and only ifλ ≥ Λ.Since the Riemann hypothesis is equivalent to the claim that all the zeroes ofH(0,z) are real, the Riemann hypothesis is equivalent to the conjecture thatΛ ≤ 0. Brad Rodgers andTerence Tao discovered the equivalence is actuallyΛ = 0 by proving zero to be the lower bound of the constant.^[18] Proving zero is also the upper bound would therefore prove the Riemann hypothesis. As of April 2020 the upper bound isΛ ≤ 0.2.^[19]

Consequences of the generalized Riemann hypothesis

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Several applications use thegeneralized Riemann hypothesis forDirichlet L-series orzeta functions of number fields rather than just the Riemann hypothesis. Many basic properties of the Riemann zeta function can easily be generalized to all Dirichlet L-series, so it is plausible that a method that proves the Riemann hypothesis for the Riemann zeta function would also work for the generalized Riemann hypothesis for Dirichlet L-functions. Several results first proved using the generalized Riemann hypothesis were later given unconditional proofs without using it, though these were usually much harder. Many of the consequences on the following list are taken fromConrad (2010).

In 1913,Grönwall showed that the generalized Riemann hypothesis implies that Gauss'slist of imaginary quadratic fields with class number 1 is complete, though Baker, Stark and Heegner later gave unconditional proofs of this without using the generalized Riemann hypothesis.
In 1917, Hardy and Littlewood showed that the generalized Riemann hypothesis implies a conjecture of Chebyshev that $\lim _{x\to 1^{-}}\sum _{p>2}(-1)^{(p+1)/2}x^{p}=+\infty ,$ which says that primes 3 mod 4 are more common than primes 1 mod 4 in some sense. (For related results, seePrime number theorem § Prime number race.)
In 1923, Hardy and Littlewood showed that the generalized Riemann hypothesis implies a weak form of theGoldbach conjecture for odd numbers: that every sufficiently large odd number is the sum of three primes, though in 1937 Vinogradov gave an unconditional proof. In 1997Deshouillers, Effinger,te Riele, and Zinoviev showed that the generalized Riemann hypothesis implies that every odd number greater than 5 is the sum of three primes. In 2013Harald Helfgott proved the ternary Goldbach conjecture without the GRH dependence, subject to some extensive calculations completed with the help of David J. Platt.
In 1934, Chowla showed that the generalized Riemann hypothesis implies that the first prime in the arithmetic progressiona modm is at mostKm²log(m)² for some fixed constantK.
In 1967, Hooley showed that the generalized Riemann hypothesis impliesArtin's conjecture on primitive roots.
In 1973, Weinberger showed that the generalized Riemann hypothesis implies that Euler's list ofidoneal numbers is complete.
Weinberger (1973) showed that the generalized Riemann hypothesis for the zeta functions of all algebraic number fields implies that any number field with class number 1 is eitherEuclidean or an imaginary quadratic number field ofdiscriminant −19, −43, −67, or −163.
In 1976, G. Miller showed that the generalized Riemann hypothesis implies that one cantest if a number is prime in polynomial time via theMiller test. In 2002, Manindra Agrawal, Neeraj Kayal and Nitin Saxena proved this result unconditionally using theAKS primality test.
Odlyzko (1990) discussed how the generalized Riemann hypothesis can be used to give sharper estimates for discriminants and class numbers of number fields.
Ono & Soundararajan (1997) showed that the generalized Riemann hypothesis implies thatRamanujan's integral quadratic formx² +y² + 10z² represents all integers that it represents locally, with exactly 18 exceptions.
In 2021, Alexander (Alex) Dunn andMaksym Radziwill provedPatterson's conjecture on cubicGauss sums, under the assumption of the GRH.^[20]^[21]

Excluded middle

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Some consequences of the RH are also consequences of its negation, and are thus theorems. In their discussion of theHecke, Deuring, Mordell, Heilbronn theorem,Ireland & Rosen (1990, p. 359) say

The method of proof here is truly amazing. If the generalized Riemann hypothesis is true, then the theorem is true. If the generalized Riemann hypothesis is false, then the theorem is true. Thus, the theorem is true!!

Care should be taken to understand what is meant by saying the generalized Riemann hypothesis is false: one should specify exactly which class of Dirichlet series has a counterexample.

Littlewood's theorem

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This concerns the sign of the error in theprime number theorem.It has been computed thatπ(x) < li(x) for allx ≤ 10²⁵ (see thistable), and no value ofx is known for whichπ(x) > li(x).

In 1914, Littlewood proved that there are arbitrarily large values ofx for which

\pi (x)>\operatorname {li} (x)+{\frac {1}{3}}{\frac {\sqrt {x}}{\log x}}\log \log \log x,

and that there are also arbitrarily large values ofx for which

\pi (x)<\operatorname {li} (x)-{\frac {1}{3}}{\frac {\sqrt {x}}{\log x}}\log \log \log x.

Thus the differenceπ(x) − li(x) changes sign infinitely many times.Skewes' number is an estimate of the value ofx corresponding to the first sign change.

Littlewood's proof is divided into two cases: the RH is assumed false (about half a page ofIngham 1932, Chapt. V), and the RH is assumed true (about a dozen pages). Stanisław Knapowski (1962) followed this up with a paper on the number of times $\Delta (n)$ changes sign in the interval $\Delta (n)$ .

Gauss's class number conjecture

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This is theconjecture (first stated in article 303 of Gauss'sDisquisitiones Arithmeticae) that there are only finitely many imaginary quadratic fields with a given class number. One way to prove it would be to show that as the discriminantD → −∞ the class numberh(D) → ∞.

The following sequence of theorems involving the Riemann hypothesis is described inIreland & Rosen 1990, pp. 358–361:

Theorem (Hecke; 1918)—LetD < 0 be the discriminant of an imaginaryquadratic number fieldK. Assume the generalized Riemann hypothesis forL-functions of all imaginary quadratic Dirichlet characters. Then there is an absolute constantC such that $h(D)>C{\frac {\sqrt {|D|}}{\log |D|}}.$

Theorem (Deuring; 1933)—If the RH is false thenh(D) > 1 if|D| is sufficiently large.

Theorem (Mordell; 1934)—If the RH is false thenh(D) → ∞ asD → −∞.

Theorem (Heilbronn; 1934)—If the generalized RH is false for theL-function of some imaginary quadratic Dirichlet character thenh(D) → ∞ asD → −∞.

(In the work of Hecke and Heilbronn, the onlyL-functions that occur are those attached to imaginary quadratic characters, and it is only for thoseL-functions thatGRH is true orGRH is false is intended; a failure of GRH for theL-function of a cubic Dirichlet character would, strictly speaking, mean GRH is false, but that was not the kind of failure of GRH that Heilbronn had in mind, so his assumption was more restricted than simplyGRH is false.)

In 1935,Carl Siegel strengthened the result without using RH or GRH in any way.^[22]^[23]

Growth of Euler's totient

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In 1983J. L. Nicolas proved that $\varphi (n)<e^{-\gamma }{\frac {n}{\log \log n}}$ for infinitely manyn, whereφ(n) isEuler's totient function andγ isEuler's constant. Ribenboim remarks that: "The method of proof is interesting, in that the inequality is shown first under the assumption that the Riemann hypothesis is true, secondly under the contrary assumption."^[24]

Generalizations and analogs

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Dirichlet L-series and other number fields

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The Riemann hypothesis can be generalized by replacing the Riemann zeta function by the formally similar, but much more general, globalL-functions. In this broader setting, one expects the non-trivial zeros of the globalL-functions to have real part 1/2. It is these conjectures, rather than the classical Riemann hypothesis only for the single Riemann zeta function, which account for the true importance of the Riemann hypothesis in mathematics.

Thegeneralized Riemann hypothesis extends the Riemann hypothesis to allDirichlet L-functions. In particular it implies the conjecture thatSiegel zeros (zeros ofL-functions between 1/2 and 1) do not exist.

Theextended Riemann hypothesis extends the Riemann hypothesis to allDedekind zeta functions ofalgebraic number fields. The extended Riemann hypothesis for abelian extension of the rationals is equivalent to the generalized Riemann hypothesis. The Riemann hypothesis can also be extended to theL-functions ofHecke characters of number fields.

Thegrand Riemann hypothesis extends it to allautomorphic zeta functions, such asMellin transforms ofHecke eigenforms.

Function fields and zeta functions of varieties over finite fields

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Artin (1924) introduced global zeta functions of (quadratic)function fields and conjectured an analogue of the Riemann hypothesis for them, which has been proved by Hasse in the genus 1 case and byWeil (1948) in general. For instance, the fact that theGauss sum, of the quadratic character of afinite field of sizeq (withq odd), has absolute value ${\sqrt {q}}$ is actually an instance of the Riemann hypothesis in the function field setting. This ledWeil (1949) to conjecture a similar statement for allalgebraic varieties; the resultingWeil conjectures were proved byPierre Deligne (1974,1980).

Arithmetic zeta functions of arithmetic schemes and their L-factors

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Arithmetic zeta functions generalise the Riemann and Dedekind zeta functions as well as the zeta functions of varieties over finite fields to every arithmetic scheme or a scheme of finite type over integers. The arithmetic zeta function of a regular connectedequidimensional arithmetic scheme of Kronecker dimensionn can be factorized into the product of appropriately defined L-factors and an auxiliary factorJean-Pierre Serre (1969–1970). Assuming a functional equation and meromorphic continuation, the generalized Riemann hypothesis for the L-factor states that its zeros inside the critical strip $\Re (s)\in (0,n)$ lie on the central line. Correspondingly, the generalized Riemann hypothesis for the arithmetic zeta function of a regular connected equidimensional arithmetic scheme states that its zeros inside the critical strip lie on vertical lines $\Re (s)=1/2,3/2,\dots ,n-1/2$ and its poles inside the critical strip lie on vertical lines $\Re (s)=1,2,\dots ,n-1$ . This is known for schemes in positive characteristic and follows fromPierre Deligne (1974,1980), but remains entirely unknown in characteristic zero.

Selberg zeta functions

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Main article:Selberg zeta function

Selberg (1956) introduced theSelberg zeta function of a Riemann surface. These are similar to the Riemann zeta function: they have a functional equation, and an infinite product similar to the Euler product but taken over closed geodesics rather than primes. TheSelberg trace formula is the analogue for these functions of theexplicit formulas in prime number theory. Selberg proved that the Selberg zeta functions satisfy the analogue of the Riemann hypothesis, with the imaginary parts of their zeros related to theeigenvalues of the Laplacian operator of the Riemann surface.

Ihara zeta functions

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TheIhara zeta function of a finite graph is an analogue of theSelberg zeta function, which was first introduced byYasutaka Ihara in the context of discrete subgroups of the two-by-two p-adic special linear group. A regular finite graph is aRamanujan graph, a mathematical model of efficient communication networks, if and only if its Ihara zeta function satisfies the analogue of the Riemann hypothesis as was pointed out byT. Sunada.

Montgomery's pair correlation conjecture

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Montgomery (1973) suggested thepair correlation conjecture that the correlation functions of the (suitably normalized) zeros of the zeta function should be the same as those of the eigenvalues of arandom hermitian matrix.Odlyzko (1987) showed that this is supported by large-scale numerical calculations of these correlation functions.

Montgomery showed that (assuming the Riemann hypothesis) at least 2/3 of all zeros are simple, and a related conjecture is that all zeros of the zeta function are simple (or more generally have no non-trivial integer linear relations between their imaginary parts).Dedekind zeta functions of algebraic number fields, which generalize the Riemann zeta function, often do have multiple complex zeros.^[25] This is because the Dedekind zeta functions factorize as a product of powers ofArtin L-functions, so zeros of Artin L-functions sometimes give rise to multiple zeros of Dedekind zeta functions. Other examples of zeta functions with multiple zeros are the L-functions of someelliptic curves: these can have multiple zeros at the real point of their critical line; theBirch-Swinnerton-Dyer conjecture predicts that the multiplicity of this zero is the rank of the elliptic curve.

Other zeta functions

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There aremany other examples of zeta functions with analogues of the Riemann hypothesis, some of which have been proved.Goss zeta functions of function fields have a Riemann hypothesis, proved bySheats (1998).The main conjecture ofIwasawa theory, proved byBarry Mazur andAndrew Wiles forcyclotomic fields, and Wiles fortotally real fields, identifies the zeros of ap-adicL-function with the eigenvalues of an operator, so can be thought of as an analogue of theHilbert–Pólya conjecture forp-adicL-functions.^[26]

Attempted proofs

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Several mathematicians have addressed the Riemann hypothesis, but none of their attempts has yet been accepted as a proof.Watkins (2021) lists some incorrect solutions.

Operator theory

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Main article:Hilbert–Pólya conjecture

Hilbert and Pólya suggested that one way to derive the Riemann hypothesis would be to find aself-adjoint operator, from the existence of which the statement on the real parts of the zeros ofζ(s) would follow when one applies the criterion on realeigenvalues. Some support for this idea comes from several analogues of the Riemann zeta functions whose zeros correspond to eigenvalues of some operator: the zeros of a zeta function of a variety over a finite field correspond to eigenvalues of aFrobenius element on anétale cohomology group, the zeros of aSelberg zeta function are eigenvalues of aLaplacian operator of a Riemann surface, and the zeros of ap-adic zeta function correspond to eigenvectors of a Galois action onideal class groups.

Odlyzko (1987) showed that the distribution of the zeros of the Riemann zeta function shares some statistical properties with the eigenvalues ofrandom matrices drawn from theGaussian unitary ensemble. This gives some support to theHilbert–Pólya conjecture.

In 1999,Michael Berry andJonathan Keating conjectured that there is some unknown quantization ${\hat {H}}$ of the classical HamiltonianH =xp so that $\zeta (1/2+i{\hat {H}})=0$ and even more strongly, that the Riemann zeros coincide with the spectrum of the operator $1/2+i{\hat {H}}$ . This is in contrast tocanonical quantization, which leads to theHeisenberg uncertainty principle $\sigma _{x}\sigma _{p}\geq {\frac {\hbar }{2}}$ and thenatural numbers as spectrum of thequantum harmonic oscillator. The crucial point is that the Hamiltonian should be a self-adjoint operator so that the quantization would be a realization of the Hilbert–Pólya program. In a connection with this quantum mechanical problem Berry and Connes had proposed that the inverse of the potential of the Hamiltonian is connected to thehalf-derivative of the function $N(s)={\frac {1}{\pi }}\operatorname {Arg} \xi (1/2+i{\sqrt {s}})$ then, in Hilbert-Polya approach $V^{-1}(x)={\sqrt {4\pi }}{\frac {d^{1/2}N(x)}{dx^{1/2}}}.$ This yields a Hamiltonian whose eigenvalues are the square of the imaginary part of the Riemann zeros, and also that thefunctional determinant of thisHamiltonian operator is just theRiemann Xi function. In fact the Riemann Xi function would be proportional to the functional determinant (Hadamard product) $\det(H+1/4+s(s-1))$ ${\frac {\xi (s)}{\xi (0)}}={\frac {\det(H+s(s-1)+1/4)}{\det(H+1/4)}}.$ However this operator is not useful in practice since it includes the inverse function (implicit function) of the potential but not the potential itself.The analogy with the Riemann hypothesis overfinite fields suggests that the Hilbert space containing eigenvectors corresponding to the zeros might be some sort of firstcohomology group of thespectrum Spec (Z) of the integers.Deninger (1998) described some of the attempts to find such a cohomology theory.^[27]

Zagier (1981) constructed a natural space of invariant functions on the upper half plane that has eigenvalues under the Laplacian operator that correspond to zeros of the Riemann zeta function—and remarked that in the unlikely event that one could show the existence of a suitable positive definite inner product on this space, the Riemann hypothesis would follow.Cartier (1982) discussed a related example, where due to a bizarre bug a computer program listed zeros of the Riemann zeta function as eigenvalues of the sameLaplacian operator.

Schumayer & Hutchinson (2011) surveyed some of the attempts to construct a suitable physical model related to the Riemann zeta function.

Lee–Yang theorem

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TheLee–Yang theorem states that the zeros of certainpartition functions instatistical mechanics all lie on a "critical line" with their real part equal to 0, and this has led to some speculation about a relationship with the Riemann hypothesis.^[28]

Turán's result

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Pál Turán (1948) showed that if the functions $\sum _{n=1}^{N}n^{-s}$ have no zeros when the real part ofs is greater than one then $T(x)=\sum _{n\leq x}{\frac {\lambda (n)}{n}}\geq 0{\text{ for }}x>0,$ where λ(n) is theLiouville function given by (−1)^r ifn hasr prime factors. He showed that this in turn would imply that the Riemann hypothesis is true. ButHaselgrove (1958) proved thatT(x) is negative for infinitely manyx (and also disproved the closely relatedPólya conjecture), andBorwein, Ferguson & Mossinghoff (2008) showed that the smallest suchx is72185376951205.Spira (1968) showed by numerical calculation that the finiteDirichlet series above forN = 19 has a zero with real part greater than 1. Turán also showed that a somewhat weaker assumption, the nonexistence of zeros with real part greater than1 +N^−1/2+ε for largeN in the finite Dirichlet series above, would also imply the Riemann hypothesis, butMontgomery (1983) showed that for all sufficiently largeN these series have zeros with real part greater than1 + (log logN)/(4 logN). Therefore, Turán's result isvacuously true and cannot help prove the Riemann hypothesis.

Noncommutative geometry

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Connes (1999,2000) has described a relationship between the Riemann hypothesis andnoncommutative geometry, and showed that a suitable analog of theSelberg trace formula for the action of theidèle class group on the adèle class space would imply the Riemann hypothesis. Some of these ideas are elaborated inLapidus (2008).

Hilbert spaces of entire functions

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Louis de Branges (1992) showed that the Riemann hypothesis would follow from a positivity condition on a certainHilbert space ofentire functions.HoweverConrey & Li (2000) showed that the necessary positivity conditions are not satisfied. Despite this obstacle, de Branges has continued to work on an attempted proof of the Riemann hypothesis along the same lines, but this has not been widely accepted by other mathematicians.^[29]

Quasicrystals

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The Riemann hypothesis implies that the zeros of the zeta function form aquasicrystal, a distribution with discrete support whoseFourier transform also has discrete support.Dyson (2009) suggested trying to prove the Riemann hypothesis by classifying, or at least studying, 1-dimensional quasicrystals.

Arithmetic zeta functions of models of elliptic curves over number fields

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When one goes from geometric dimension one, e.g. analgebraic number field, to geometric dimension two, e.g. a regular model of anelliptic curve over a number field, the two-dimensional part of the generalized Riemann hypothesis for thearithmetic zeta function of the model deals with the poles of the zeta function. In dimension one the study of the zeta integral inTate's thesis does not lead to new important information on the Riemann hypothesis. Contrary to this, in dimension two work ofIvan Fesenko on two-dimensional generalisation of Tate's thesis includes an integral representation of a zeta integral closely related to the zeta function. In this new situation, not possible in dimension one, the poles of the zeta function can be studied via the zeta integral and associated adele groups. Related conjecture ofFesenko (2010) on the positivity of the fourth derivative of a boundary function associated to the zeta integral essentially implies the pole part of the generalized Riemann hypothesis. Suzuki (2011) proved that the latter, together with some technical assumptions, implies Fesenko's conjecture.

Multiple zeta functions

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Deligne's proof of the Riemann hypothesis over finite fields used the zeta functions of product varieties, whose zeros and poles correspond to sums of zeros and poles of the original zeta function, in order to bound the real parts of the zeros of the original zeta function. By analogy,Kurokawa (1992) introduced multiple zeta functions whose zeros and poles correspond to sums of zeros and poles of the Riemann zeta function. To make the series converge he restricted to sums of zeros or poles all with non-negative imaginary part. So far, the known bounds on the zeros and poles of the multiple zeta functions are not strong enough to give useful estimates for the zeros of the Riemann zeta function.

Location of the zeros

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Number of zeros

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The functional equation combined with theargument principle implies that the number of zeros of the zeta function with imaginary part between 0 andT is given by

N(T)={\frac {1}{\pi }}\mathop {\mathrm {Arg} } (\xi (s))={\frac {1}{\pi }}\mathop {\mathrm {Arg} } (\Gamma ({\tfrac {s}{2}})\pi ^{-{\frac {s}{2}}}\zeta (s)s(s-1)/2)

fors = 1/2 +iT, where the argument is defined by varying it continuously along the line withIm(s) =T, starting with argument 0 at∞ +iT. This is the sum of a large but well understood term

{\frac {1}{\pi }}\mathop {\mathrm {Arg} } (\Gamma ({\tfrac {s}{2}})\pi ^{-s/2}s(s-1)/2)={\frac {T}{2\pi }}\log {\frac {T}{2\pi }}-{\frac {T}{2\pi }}+7/8+O(1/T)

and a small but rather mysterious term

S(T)={\frac {1}{\pi }}\mathop {\mathrm {Arg} } (\zeta (1/2+iT))=O(\log T).

So the density of zeros with imaginary part nearT is about log(T)/(2π), and the functionS describes the small deviations from this. The functionS(t) jumps by 1 at each zero of the zeta function, and fort ≥ 8 it decreasesmonotonically between zeros with derivative close to−logt.

Trudgian (2014) proved that, ifT >e, then

|N(T)-{\frac {T}{2\pi }}\log {\frac {T}{2\pi e}}|\leq 0.112\log T+0.278\log \log T+3.385+{\frac {0.2}{T}}

Karatsuba (1996) proved that every interval(T,T +H] for $H\geq T^{{\frac {27}{82}}+\varepsilon }$ contains at least

H(\log T)^{\frac {1}{3}}e^{-c{\sqrt {\log \log T}}}

points where the functionS(t) changes sign.

Selberg (1946) showed that the average moments of even powers ofS are given by

\int _{0}^{T}|S(t)|^{2k}dt={\frac {(2k)!}{k!(2\pi )^{2k}}}T(\log \log T)^{k}+O(T(\log \log T)^{k-1/2}).

This suggests thatS(T)/(log logT)^1/2 resembles aGaussian random variable with mean 0 and variance 2π² (Ghosh (1983) proved this fact).In particular |S(T)| is usually somewhere around (log logT)^1/2, but occasionally much larger. The exact order of growth ofS(T) is not known. There has been no unconditional improvement to Riemann's original boundS(T) =O(logT), though the Riemann hypothesis implies the slightly smaller boundS(T) =O(logT/log logT).^[15] The true order of magnitude may be somewhat less than this, as random functions with the same distribution asS(T) tend to have growth of order about log(T)^1/2. In the other direction it cannot be too small:Selberg (1946) showed thatS(T) ≠o((logT)^1/3/(log logT)^7/3), and assuming the Riemann hypothesis Montgomery showed thatS(T) ≠o((logT)^1/2/(log logT)^1/2).

Numerical calculations confirm thatS grows very slowly:|S(T)| < 1 forT < 280,|S(T)| < 2 forT <6800000, and the largest value of |S(T)| found so far is not much larger than 3.^[30]

Riemann's estimateS(T) =O(logT) implies that the gaps between zeros are bounded, and Littlewood improved this slightly, showing that the gaps between their imaginary parts tend to 0.

Theorem of Hadamard and de la Vallée-Poussin

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Hadamard (1896) andde la Vallée-Poussin (1896) independently proved that no zeros could lie on the lineRe(s) = 1. Together with the functional equation and the fact that there are no zeros with real part greater than 1, this showed that all non-trivial zeros must lie in the interior of the critical strip0 < Re(s) < 1. This was a key step in their first proofs of theprime number theorem.

Both the original proofs that the zeta function has no zeros with real part 1 are similar, and depend on showing that ifζ(1 +it) vanishes, thenζ(1 + 2it) is singular, which is not possible. One way of doing this is by using the inequality

|\zeta (\sigma )^{3}\zeta (\sigma +it)^{4}\zeta (\sigma +2it)|\geq 1

forσ > 1,t real, and looking at the limit asσ → 1. This inequality follows by taking the real part of the log of the Euler product to see that

|\zeta (\sigma +it)|=\exp \Re \sum _{p^{n}}{\frac {p^{-n(\sigma +it)}}{n}}=\exp \sum _{p^{n}}{\frac {p^{-n\sigma }\cos(t\log p^{n})}{n}},

where the sum is over all prime powerspⁿ, so that

|\zeta (\sigma )^{3}\zeta (\sigma +it)^{4}\zeta (\sigma +2it)|=\exp \sum _{p^{n}}p^{-n\sigma }{\frac {3+4\cos(t\log p^{n})+\cos(2t\log p^{n})}{n}}

which is at least 1 because all the terms in the sum are positive, due to the inequality

3+4\cos(\theta )+\cos(2\theta )=2(1+\cos(\theta ))^{2}\geq 0.

Zero-free regions

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The most extensive computer search by Platt andTrudgian^[19] for counterexamples of the Riemann hypothesis has verified it for|t| ≤3.0001753328×10¹². Beyond that zero-free regions are known as inequalities concerningσ +i t, which can be zeroes. The oldest version is fromDe la Vallée-Poussin (1899–1900), who proved there is a region without zeroes that satisfies1 −σ ≥⁠C/log(t)⁠ for some positive constantC. In other words, zeros cannot be too close to the lineσ = 1: there is a zero-free region close to this line. This has been enlarged by several authors using methods such asVinogradov's mean-value theorem.

The most recent paper^[31] by Mossinghoff, Trudgian and Yang is from December 2022 and provides four zero-free regions that improved the previous results of Kevin Ford from 2002, Mossinghoff and Trudgian themselves from 2015 and Pace Nielsen's slight improvement of Ford from October 2022:

\sigma \geq 1-{\frac {1}{5.558691\log |t|}}

whenever

|t|\geq 2

\sigma \geq 1-{\frac {1}{55.241(\log {|t|})^{2/3}(\log {\log {|t|}})^{1/3}}}

whenever

|t|\geq 3

(largest known region in the bound

3.0001753328\cdot 10^{12}\leq |t|\leq \exp(64.1)\approx 6.89\cdot 10^{27}

\sigma \geq 1-{\frac {0.04962-{\frac {0.0196}{1.15+\log 3+{\frac {1}{6}}\log t+\log \log t}}}{0.685+\log 3+{\frac {1}{6}}\log t+1.155\cdot \log \log t}}

whenever

|t|\geq 1.88\cdot 10^{14}

(largest known region in the bound

\exp(64.1)\leq |t|\leq \exp(1000)\approx 1.97\cdot 10^{434}

) and

\sigma \geq 1-{\frac {0.05035}{{\frac {27}{164}}(\log {|t|})+7.096}}+{\frac {0.0349}{({\frac {27}{164}}(\log {|t|})+7.096)^{2}}}

whenever

|t|\geq \exp(1000)

(largest known region in its own bound)

The paper also presents an improvement to the second zero-free region, whose bounds are unknown on account of $|t|$ being merely assumed to be "sufficiently large" to fulfill the requirements of the paper's proof. This region is

\sigma \geq 1-{\frac {1}{48.1588(\log {|t|})^{2/3}(\log {\log {|t|}})^{1/3}}}

Zeros on the critical line

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Hardy (1914) andHardy & Littlewood (1921) showed there are infinitely many zeros on the critical line, by considering moments of certain functions related to the zeta function.Selberg (1942) proved that at least a (small) positive proportion of zeros lie on the line.Levinson (1974) improved this to one-third of the zeros by relating the zeros of the zeta function to those of its derivative, andConrey (1989) improved this further to two-fifths. In 2020, this estimate was extended to five-twelfths by Pratt, Robles,Zaharescu and Zeindler^[32] by considering extended mollifiers that can accommodate higher order derivatives of the zeta function and their associated Kloosterman sums.

Most zeros lie close to the critical line. More precisely,Bohr & Landau (1914) showed that for any positiveε, the number of zeros with real part at least 1/2+ε and imaginary part at between −T andT is $O(T)$ . Combined with the facts that zeros on the critical strip are symmetric about the critical line and that the total number of zeros in the critical strip is $\Theta (T\log T)$ ,almost all non-trivial zeros are within a distanceε of the critical line.Ivić (1985) gives several more precise versions of this result, calledzero density estimates, which bound the number of zeros in regions with imaginary part at mostT and real part at least1/2 +ε.

Hardy–Littlewood conjectures

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In 1914Godfrey Harold Hardy proved that $\zeta \left({\tfrac {1}{2}}+it\right)$ has infinitely many real zeros.

The next two conjectures ofHardy andJohn Edensor Littlewood on the distance between real zeros of $\zeta \left({\tfrac {1}{2}}+it\right)$ and on the density of zeros of $\zeta \left({\tfrac {1}{2}}+it\right)$ on the interval $(T,T+H]$ for sufficiently large $T>0$ , and $H=T^{a+\varepsilon }$ and with as small as possible value of $a>0$ , where $\varepsilon >0$ is an arbitrarily small number, open two new directions in the investigation of the Riemann zeta function:

For any $\varepsilon >0$ there exists a lower bound $T_{0}=T_{0}(\varepsilon )>0$ such that for $T\geq T_{0}$ and $H=T^{{\tfrac {1}{4}}+\varepsilon }$ the interval $(T,T+H]$ contains a zero of odd order of the function $\zeta {\bigl (}{\tfrac {1}{2}}+it{\bigr )}$ .

Let $N(T)$ be the total number of real zeros, and $N_{0}(T)$ be the total number of zeros of odd order of the function $~\zeta \left({\tfrac {1}{2}}+it\right)~$ lying on the interval $(0,T]~$ .

For any $\varepsilon >0$ there exists $T_{0}=T_{0}(\varepsilon )>0$ and some $c=c(\varepsilon )>0$ , such that for $T\geq T_{0}$ and $H=T^{{\tfrac {1}{2}}+\varepsilon }$ the inequality $N_{0}(T+H)-N_{0}(T)\geq cH$ is true.

Selberg's zeta function conjecture

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Main article:Selberg's zeta function conjecture

Atle Selberg (1942) investigated the problem of Hardy–Littlewood2 and proved that for anyε > 0 there exists such $T_{0}=T_{0}(\varepsilon )>0$ andc =c(ε) > 0, such that for $T\geq T_{0}$ and $H=T^{0.5+\varepsilon }$ the inequality $N(T+H)-N(T)\geq cH\log T$ is true. Selberg conjectured that this could be tightened to $H=T^{0.5}$ .A. A. Karatsuba (1984a,1984b,1985) proved that for a fixedε satisfying the condition 0 <ε < 0.001, a sufficiently largeT and $H=T^{a+\varepsilon }$ , $a={\tfrac {27}{82}}={\tfrac {1}{3}}-{\tfrac {1}{246}}$ , the interval(T,T+H) contains at leastcH log(T) real zeros of theRiemann zeta function $\zeta \left({\tfrac {1}{2}}+it\right)$ and therefore confirmed the Selberg conjecture. The estimates of Selberg and Karatsuba can not be improved in respect of the order of growth asT → ∞.

Karatsuba (1992) proved that an analog of the Selberg conjecture holds for almost all intervals(T,T+H], $H=T^{\varepsilon }$ , whereε is an arbitrarily small fixed positive number. The Karatsuba method permits to investigate zeros of the Riemann zeta function on "supershort" intervals of the critical line, that is, on the intervals(T,T+H], the lengthH of which grows slower than any, even arbitrarily small degreeT. In particular, he proved that for any given numbersε, $\varepsilon _{1}$ satisfying the conditions $0<\varepsilon ,\varepsilon _{1}<1$ almost all intervals(T,T+H] for $H\geq \exp {\{(\log T)^{\varepsilon }\}}$ contain at least $H(\log T)^{1-\varepsilon _{1}}$ zeros of the function $\zeta \left({\tfrac {1}{2}}+it\right)$ . This estimate is quite close to the one that follows from the Riemann hypothesis.

Numerical calculations

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The function

\pi ^{-{\frac {s}{2}}}\Gamma ({\tfrac {s}{2}})\zeta (s)

has the same zeros as the zeta function in the critical strip, and is real on the critical line because of the functional equation, so one can prove the existence of zeros exactly on the real line between two points by checking numerically that the function has opposite signs at these points. Usually one writes

\zeta ({\tfrac {1}{2}}+it)=Z(t)e^{-i\theta (t)}

where Hardy'sZ function and theRiemann–Siegel theta functionθ are uniquely defined by this and the condition that they are smooth real functions withθ(0) = 0.By finding many intervals where the functionZ changes sign one can show that there are many zeros on the critical line. To verify the Riemann hypothesis up to a givenimaginary partT of the zeros, one also has to check that there are no further zeros off the line in this region. This can be done by calculating the total number of zeros in the region usingTuring's method and checking that it is the same as the number of zeros found on the line. This allows one to verify the Riemann hypothesis computationally up to any desired value ofT (provided all the zeros of the zeta function in this region are simple and on the critical line).^[33]^[34]

These calculations can also be used to estimate $\pi (x)$ for finite ranges of $x {\displaystyle x}$ . For example, using the latest result from 2020 (zeros up to height $3\times 10^{12}$ ), it has been shown that

|\pi (x)-\operatorname {li} (x)|<{\frac {1}{8\pi }}{\sqrt {x}}\log(x),\qquad {\text{for }}2657\leq x\leq 1.101\times 10^{26}.

In general, this inequality holds if

x\geq 2657

and

{\frac {9.06}{\log {\log {x}}}}{\sqrt {\frac {x}{\log {x}}}}\leq T,

where $T {\displaystyle T}$ is the largest known value such that the Riemann hypothesis is true for all zeros $\rho$ with $\Im {\left(\rho \right)}\in \left(0,T\right]$ .^[35]

Some calculations of zeros of the zeta function are listed below, where the "height" of a zero is the magnitude of its imaginary part, and the height of thenth zero is denoted byγ_n. So far all zeros that have been checked are on the critical line and are simple. (A multiple zero would cause problems for the zero finding algorithms, which depend on finding sign changes between zeros.) For tables of the zeros, seeHaselgrove & Miller (1960) orOdlyzko.

Year	Number of zeros	Author
1859?	3	B. Riemann used theRiemann–Siegel formula (unpublished, but reported inSiegel 1932).
1903	15	J. P.Gram (1903) used theEuler–Maclaurin formula and discoveredGram's law. He showed that all 10 zeros with imaginary part at most 50 range lie on the critical line with real part 1/2 by computing the sum of the inverse 10th powers of the roots he found.
1914	79 (γ_n ≤ 200)	R. J.Backlund (1914) introduced a better method of checking all the zeros up to that point are on the line, by studying the argumentS(T) of the zeta function.
1925	138 (γ_n ≤ 300)	J. I.Hutchinson (1925) found the first failure of Gram's law, at the Gram pointg₁₂₆.
1935	195	E. C.Titchmarsh (1935) used the recently rediscoveredRiemann–Siegel formula, which is much faster than Euler–Maclaurin summation. It takes about O(T^3/2 + ε) steps to check zeros with imaginary part less thanT, while the Euler–Maclaurin method takes about O(T^2 + ε) steps.
1936	1041	E. C.Titchmarsh (1936) and L. J. Comrie were the last to find zeros by hand.
1953	1104	A. M.Turing (1953) found a more efficient way to check that all zeros up to some point are accounted for by the zeros on the line, by checking thatZ has the correct sign at several consecutive Gram points and using the fact thatS(T) has average value 0. This requires almost no extra work because the sign ofZ at Gram points is already known from finding the zeros, and is still the usual method used. This was the first use of a digital computer to calculate the zeros.
1956	15000	D. H.Lehmer (1956) discovered a few cases where the zeta function has zeros that are "only just" on the line: two zeros of the zeta function are so close together that it is unusually difficult to find a sign change between them. This is called "Lehmer's phenomenon", and first occurs at the zeros with imaginary parts 7005.063 and 7005.101, which differ by only .04 while the average gap between other zeros near this point is about 1.
1956	25000	D. H. Lehmer
1958	35337	N. A. Meller
1966	250000	R. S. Lehman
1968	3500000	Rosser, Yohe & Schoenfeld (1969) stated Rosser's rule (described below).
1977	40000000	R. P. Brent
1979	81000001	R. P. Brent
1982	200000001	R. P. Brent,J. van de Lune,H. J. J. te Riele, D. T. Winter
1983	300000001	J. van de Lune, H. J. J. te Riele
1986	1500000001	van de Lune, te Riele & Winter (1986) gave some statistical data about the zeros and give several graphs ofZ at places where it has unusual behavior.
1987	A few of large (≈10¹²) height	A. M. Odlyzko (1987) computed smaller numbers of zeros of much larger height, around 10¹², to high precision to checkMontgomery's pair correlation conjecture.
1992	A few of large (≈10²⁰) height	A. M. Odlyzko (1992) computed 175 million zeros of heights around 10²⁰ and a few more of heights around 2×10²⁰, and gave an extensive discussion of the results.
1998	10000 of large (≈10²¹) height	A. M. Odlyzko (1998) computed some zeros of height about 10²¹
2001	10¹⁰	J. van de Lune (unpublished)
2004	≈9×10¹¹^[36]	S. Wedeniwski (ZetaGrid distributed computing)
2004	10¹³ and a few of large (up to ≈10²⁴) heights	XavierGourdon (2004) and Patrick Demichel used theOdlyzko–Schönhage algorithm. They also checked two billion zeros around heightsγ_n = 10¹³, 10¹⁴, ..., 10²⁴.
2020	1.2363×10¹³ (γ_n ≤ 3×10¹²)	Platt & Trudgian (2021). They also verified the work ofGourdon (2004) and others.

Gram points

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AGram point is a point on the critical line 1/2 + it where the zeta function is real and non-zero. Using the expression for the zeta function on the critical line,ζ(1/2 +it) =Z(t)e^−iθ(t), where Hardy's function,Z, is real for realt, andθ is theRiemann–Siegel theta function, we see that zeta is real whensin(θ(t)) = 0. This implies thatθ(t) is an integer multiple ofπ, which allows for the location of Gram points to be calculated fairly easily by inverting the formula forθ. They are usually numbered asg_n forn = 0, 1, ..., whereg_n is the unique solution ofθ(t) =nπ.

Gram observed that there was often exactly one zero of the zeta function between any two consecutive Gram points; Hutchinson called this observationGram's law. There are several other closely related statements that are also sometimes called Gram's law: for example,(−1)ⁿZ(g_n) is usually positive, orZ(t) usually has opposite sign at consecutive Gram points. The imaginary partsγ_n of the first few zeros (in blue) and the first few Gram pointsg_n are given in the following table

		g₋₁	γ₁	g₀	γ₂	g₁	γ₃	g₂	γ₄	g₃	γ₅	g₄	γ₆	g₅
0	3.436	9.667	14.135	17.846	21.022	23.170	25.011	27.670	30.425	31.718	32.935	35.467	37.586	38.999

This is a polar plot of the first 20 real valuesr_n of the zeta function along the critical line,ζ(1/2 +it), witht running from 0 to 50. The values ofr_n in this range are the first 10 non-trivialRiemann zeta function zeros and the first 10Gram points, each labeled byn. Fifty red points have been plotted between eachr_n, and the zeros are projected onto concentric magenta rings scaled to show the relative distance between their values of t. Gram's law states that the curve usually crosses the real axis once between zeros.

The first failure of Gram's law occurs at the 127th zero and the Gram pointg₁₂₆, which are in the "wrong" order.

g₁₂₄	γ₁₂₆	g₁₂₅	g₁₂₆	γ₁₂₇	γ₁₂₈	g₁₂₇	γ₁₂₉	g₁₂₈
279.148	279.229	280.802	282.455	282.465	283.211	284.104	284.836	285.752

A Gram pointt is called good if the zeta function is positive at1/2 +it. The indices of the "bad" Gram points whereZ has the "wrong" sign are 126, 134, 195, 211, ... (sequenceA114856 in theOEIS). AGram block is an interval bounded by two good Gram points such that all the Gram points between them are bad. A refinement of Gram's law called Rosser's rule due toRosser, Yohe & Schoenfeld (1969) says that Gram blocks often have the expected number of zeros in them (the same as the number of Gram intervals), even though some of the individual Gram intervals in the block may not have exactly one zero in them. For example, the interval bounded byg₁₂₅ andg₁₂₇ is a Gram block containing a unique bad Gram pointg₁₂₆, and contains the expected number 2 of zeros although neither of its two Gram intervals contains a unique zero. Rosser et al. checked that there were no exceptions to Rosser's rule in the first 3 million zeros, although there are infinitely many exceptions to Rosser's rule over the entire zeta function.

Gram's rule and Rosser's rule both say that in some sense zeros do not stray too far from their expected positions. The distance of a zero from its expected position is controlled by the functionS defined above, which grows extremely slowly: its average value is of the order of (log logT)^1/2, which only reaches 2 for T around 10²⁴. This means that both rules hold most of the time for smallT but eventually break down often. Indeed,Trudgian (2011) showed that both Gram's law and Rosser's rule fail in a positive proportion of cases. To be specific, it is expected that in about 66% one zero is enclosed by two successive Gram points, but in 17% no zero and in 17% two zeros are in such a Gram-interval on the long runHanga (2020).

Arguments for and against the Riemann hypothesis

[edit]

Mathematical papers about the Riemann hypothesis tend to be cautiously noncommittal about its truth. Of authors who express an opinion, most of them, such asRiemann (1859) andBombieri (2000), imply that they expect (or at least hope) that it is true. The few authors who express serious doubt about it includeIvić (2008), who lists some reasons for skepticism, andLittlewood (1962), who flatly states that he believes it false, that there is no evidence for it and no imaginable reason it would be true. The consensus of the survey articles (Bombieri 2000,Conrey 2003, andSarnak 2005) is that the evidence for it is strong but not overwhelming, so that while it is probably true there is reasonable doubt.

Some of the arguments for and against the Riemann hypothesis are listed bySarnak (2005),Conrey (2003), andIvić (2008), and include the following:

Several analogues of the Riemann hypothesis have already been proved. The proof of the Riemann hypothesis for varieties over finite fields byDeligne (1974) is possibly the single strongest theoretical reason in favor of the Riemann hypothesis. This provides some evidence for the more general conjecture that all zeta functions associated withautomorphic forms satisfy a Riemann hypothesis, which includes the classical Riemann hypothesis as a special case. SimilarlySelberg zeta functions satisfy the analogue of the Riemann hypothesis, and are in some ways similar to the Riemann zeta function, having a functional equation and an infinite product expansion analogous to the Euler product expansion. But there are also some major differences; for example, they are not given by Dirichlet series. The Riemann hypothesis for theGoss zeta function was proved bySheats (1998). In contrast to these positive examples, someEpstein zeta functions do not satisfy the Riemann hypothesis even though they have an infinite number of zeros on the critical line.^[15] These functions are quite similar to the Riemann zeta function, and have a Dirichlet series expansion and afunctional equation, but the ones known to fail the Riemann hypothesis do not have anEuler product and are not directly related toautomorphic representations.
At first, the numerical verification that many zeros lie on the line seems strong evidence for it. But analytic number theory has had many conjectures supported by substantial numerical evidence that turned out to be false. SeeSkewes number for a notorious example, where the first exception to a plausible conjecture related to the Riemann hypothesis probably occurs around 10³¹⁶; a counterexample to the Riemann hypothesis with imaginary part this size would be far beyond anything that can currently be computed using a direct approach. The problem is that the behavior is often influenced by very slowly increasing functions such as log logT, that tend to infinity, but do so so slowly that this cannot be detected by computation. Such functions occur in the theory of the zeta function controlling the behavior of its zeros; for example the functionS(T) above has average size around (log logT)^1/2. AsS(T) jumps by at least 2 at any counterexample to the Riemann hypothesis, one might expect any counterexamples to the Riemann hypothesis to start appearing only whenS(T) becomes large. It is never much more than 3 as far as it has been calculated, but is known to be unbounded, suggesting that calculations may not have yet reached the region of typical behavior of the zeta function.
Denjoy's probabilistic argument for the Riemann hypothesis^[37] is based on the observation that ifμ(x) is a random sequence of "1"s and "−1"s then, for everyε > 0, thepartial sums $M(x)=\sum _{n\leq x}\mu (n)$ (the values of which are positions in asimple random walk) satisfy the bound $M(x)=O(x^{1/2+\varepsilon })$ withprobability 1. The Riemann hypothesis is equivalent to this bound for theMöbius function μ and theMertens functionM derived in the same way from it. In other words, the Riemann hypothesis is in some sense equivalent to saying thatμ(x) behaves like a random sequence of coin tosses. Whenμ(x) is nonzero its sign gives theparity of the number of prime factors ofx, so informally the Riemann hypothesis says that the parity of the number of prime factors of an integer behaves randomly. Such probabilistic arguments in number theory often give the right answer, but tend to be very hard to make rigorous, and occasionally give the wrong answer for some results, such asMaier's theorem.
The calculations inOdlyzko (1987) show that the zeros of the zeta function behave very much like the eigenvalues of a randomHermitian matrix, suggesting that they are the eigenvalues of some self-adjoint operator, which would imply the Riemann hypothesis. All attempts to find such an operator have failed.
There are several theorems, such asGoldbach's weak conjecture for sufficiently large odd numbers, that were first proved using the generalized Riemann hypothesis, and later shown to be true unconditionally. This could be considered as weak evidence for the generalized Riemann hypothesis, as several of its "predictions" are true.
Lehmer's phenomenon,^[38] where two zeros are sometimes very close, is sometimes given as a reason to disbelieve the Riemann hypothesis. But one would expect this to happen occasionally by chance even if the Riemann hypothesis is true, and Odlyzko's calculations suggest that nearby pairs of zeros occur just as often as predicted byMontgomery's conjecture.
Patterson suggests that the most compelling reason for the Riemann hypothesis for most mathematicians is the hope that primes are distributed as regularly as possible.^[39]

Notes

[edit]

^Bombieri (2000).
^Euler, Leonhard (1744).Variae observationes circa series infinitas.Commentarii academiae scientiarum Petropolitanae 9, pp. 160–188, Theorems 7 and 8. In Theorem 7 Euler proves the formula in the special case $s=1$ , and in Theorem 8 he proves it more generally. In the first corollary to his Theorem 7 he notes that $\zeta (1)=\log \infty$ , and he makes use of this latter result in his Theorem 19, to show that the sum of the inverses of the prime numbers is $\log \log \infty$ .
^Values forζ can be found by calculating, e.g.,ζ(1/2 − 30i).("Wolframalpha computational intelligence".wolframalpha.com. Wolfram. Retrieved2 October 2022.
^Ingham (1932), Theorem 30, p. 83;Montgomery & Vaughan (2007), p. 430.
^Ingham (1932), p. 82.
^von Koch, Niels Helge (1901)."Sur la distribution des nombres premiers".Acta Mathematica.24:159–182.doi:10.1007/BF02403071.S2CID 119914826.
^Dudek, Adrian W. (2014). "On the Riemann hypothesis and the difference between primes".International Journal of Number Theory.11 (3):771–778.arXiv:1402.6417.Bibcode:2014arXiv1402.6417D.doi:10.1142/S1793042115500426.ISSN 1793-0421.S2CID 119321107.
^Landau, Edmund (1924), "Über die Möbiussche Funktion",Rend. Circ. Mat. Palermo,48 (2):277–280,doi:10.1007/BF03014702,S2CID 123636883
^Titchmarsh, Edward Charles (1927), "A consequence of the Riemann hypothesis",J. London Math. Soc.,2 (4):247–254,doi:10.1112/jlms/s1-2.4.247
^Maier, Helmut; Montgomery, Hugh (2009), "The sum of the Möbius function",Bull. London Math. Soc.,41 (2):213–226,doi:10.1112/blms/bdn119,hdl:2027.42/135214,S2CID 121272525
^Soundararajan, Kannan (2009), "Partial sums of the Möbius function",J. Reine Angew. Math.,2009 (631):141–152,arXiv:0705.0723,doi:10.1515/CRELLE.2009.044,S2CID 16501321
^Robin (1984).
^Lagarias, Jeffrey C. (2002), "An elementary problem equivalent to the Riemann hypothesis",The American Mathematical Monthly,109 (6):534–543,arXiv:math/0008177,doi:10.2307/2695443,ISSN 0002-9890,JSTOR 2695443,MR 1908008,S2CID 15884740
^Broughan (2017), Corollary 5.35.
^^a ^b ^cTitchmarsh (1986).
^Nicely (1999).
^Baez-Duarte, Luis (2005)."A general strong Nyman-Beurling criterion for the Riemann hypothesis".Publications de l'Institut Mathématique. Nouvelle Série.78 (92):117–125.arXiv:math/0505453.doi:10.2298/PIM0578117B.S2CID 17406178.
^Rodgers & Tao (2020).
^^a ^bPlatt & Trudgian (2021).
^"Caltech Mathematicians Solve 19th Century Number Riddle".California Institute of Technology. October 31, 2022.
^Dunn, Alexander;Radziwiłł, Maksym (2021). "Bias in cubic Gauss sums: Patterson's conjecture".arXiv:2109.07463 [math.NT].
^Goldfeld, Dorian (1985)."Gauss' class number problem for imaginary quadratic fields".Bulletin of the American Mathematical Society.13 (1):23–37.doi:10.1090/S0273-0979-1985-15352-2.ISSN 0273-0979.
^Siegel, Carl (1935)."Über die Classenzahl quadratischer Zahlkörper".Acta Arithmetica.1 (1):83–86.doi:10.4064/aa-1-1-83-86.ISSN 0065-1036. Retrieved8 April 2024.
^Ribenboim (1996), p. 320.
^Radziejewski (2007).
^Wiles (2000).
^Leichtnam (2005).
^Knauf (1999).
^Sarnak (2005).
^Odlyzko (2002).
^Mossinghoff, Michael J.; Trudgian, Timothy S.; Yang, Andrew (2022-12-13). "Explicit zero-free regions for the Riemann zeta-function".arXiv:2212.06867 [math.NT].
^Pratt, Kyle; Robles, Nicolas; Zaharescu, Alexandru; Zeindler, Dirk (2020). "More than five-twelfths of the zeros ofζ are on the critical line".Res Math Sci.7.arXiv:1802.10521.doi:10.1007/s40687-019-0199-8.S2CID 202542332.
^Hejhal, Dennis A.; Odlyzko, Andrew M."Alan Turing and the Riemann Zeta Function". University of Minnesota.
^Yu, Matiyasevich (2020)."The Riemann Hypothesis in computer science".Theoretical Computer Science.807:257–265.doi:10.1016/j.tcs.2019.07.028.
^Johnston, Daniel R. (29 July 2022)."Improving bounds on prime counting functions by partial verification of the Riemann hypothesis".The Ramanujan Journal.59 (4):1307–1321.arXiv:2109.02249.doi:10.1007/s11139-022-00616-x.S2CID 237420836.
^Weisstein, Eric W.,"Riemann Zeta Function Zeros",MathWorld: "ZetaGrid is a distributed computing project attempting to calculate as many zeros as possible. It had reached 1029.9 billion zeros as of Feb. 18, 2005."
^Edwards (1974).
^Lehmer (1956).
^p. 75: "One should probably add to this list the 'Platonic' reason that one expects the natural numbers to be the most perfect idea conceivable, and that this is only compatible with the primes being distributed in the most regular fashion possible ..."

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Popular expositions

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Sabbagh, Karl (2003a),The greatest unsolved problem in mathematics, Farrar, Straus and Giroux, New York,ISBN 978-0-374-25007-2,MR 1979664
Sabbagh, Karl (2003b),Dr. Riemann's zeros, Atlantic Books, London,ISBN 978-1-843-54101-1
du Sautoy, Marcus (2003),The music of the primes, HarperCollins Publishers,ISBN 978-0-06-621070-4,MR 2060134
Rockmore, Dan (2005),Stalking the Riemann hypothesis, Pantheon Books,ISBN 978-0-375-42136-5,MR 2269393
Derbyshire, John (2003),Prime Obsession, Joseph Henry Press, Washington, DC,ISBN 978-0-309-08549-6,MR 1968857
Watkins, Matthew (2015),Mystery of the Prime Numbers, Liberalis Books,ISBN 978-1782797814,MR 0000000
Frenkel, Edward (2014),The Riemann Hypothesis Numberphile, Mar 11, 2014 (video)
Nahin, Paul J. (2021).In Pursuit of Zeta-3: The World's Most Mysterious Unsolved Math Problem. Princeton University Press.ISBN 978-0691206073.

Note: Derbyshire 2003, Rockmore 2005, Sabbagh 2003a, Sabbagh 2003b, Sautoy 2003, and Watkins 2015 are non-technical. Edwards 1974, Patterson 1988, Borwein/Choi/Rooney/Weirathmueller 2008, Mazur/Stein 2015, Broughan 2017, and Nahin 2021 give mathematical introductions. Titchmarsh 1986, Ivić 1985, and Karatsuba/Voronin 1992 are advancedmonographs.

External links

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Media related toRiemann hypothesis at Wikimedia Commons

American Institute of Mathematics,Riemann hypothesis
Zeroes database, 103 800 788 359 zeroes
Apostol, Tom,Where are the zeros of zeta of s? Poem about the Riemann hypothesis,sung byJohn Derbyshire.
Borwein, Peter,The Riemann Hypothesis(PDF), archived fromthe original(PDF) on 2009-03-27 (Slides for a lecture)
Conrad, K. (2010),Consequences of the Riemann hypothesis
Conrey, J. Brian; Farmer, David W,Equivalences to the Riemann hypothesis, archived fromthe original on 2010-03-16
Gourdon, Xavier; Sebah, Pascal (2004),Computation of zeros of the Zeta function (Reviews the GUE hypothesis, provides an extensive bibliography as well).
Odlyzko, Andrew,Home page includingpapers on the zeros of the zeta function andtables of the zeros of the zeta function
Odlyzko, Andrew (2002),Zeros of the Riemann zeta function: Conjectures and computations(PDF) Slides of a talk
Pegg, Ed (2004),Ten Trillion Zeta Zeros, Math Games website, archived fromthe original on 2004-11-02, retrieved2004-10-20. A discussion of Xavier Gourdon's calculation of the first ten trillion non-trivial zeros
Rubinstein, Michael,algorithm for generating the zeros, archived fromthe original on 2007-04-27.
du Sautoy, Marcus (2006),Prime Numbers Get Hitched, Seed Magazine, archived from the original on 2017-09-22, retrieved2006-03-27
Watkins, Matthew R. (2021-02-27),Proposed (dis)proofs of the Riemann Hypothesis,archived from the original on December 9, 2022
Zetagrid (2002) A distributed computing project that attempted to disprove Riemann's hypothesis; closed in November 2005

v t e L-functions innumber theory
Analytic examples	Riemann zeta function DirichletL-functions L-functions of Hecke characters AutomorphicL-functions Selberg class
Algebraic examples	Dedekind zeta functions ArtinL-functions Hasse–WeilL-functions MotivicL-functions
Theorems	Analytic class number formula Riemann–von Mangoldt formula Weil conjectures
Analytic conjectures	Riemann hypothesis Generalized Riemann hypothesis Lindelöf hypothesis Ramanujan–Petersson conjecture Artin conjecture
Algebraic conjectures	Birch and Swinnerton-Dyer conjecture Deligne's conjecture Beilinson conjectures Bloch–Kato conjecture Langlands conjecture
p-adicL-functions	Main conjecture of Iwasawa theory Selmer group Euler system

v t e Bernhard Riemann
Cauchy–Riemann equations Generalized Riemann hypothesis Grand Riemann hypothesis Grothendieck–Hirzebruch–Riemann–Roch theorem Hirzebruch–Riemann–Roch theorem Local zeta function Measurable Riemann mapping theorem Riemann (crater) Riemann Xi function Riemann curvature tensor Riemann hypothesis Riemann integral Riemann invariant Riemann mapping theorem Riemann form Riemann problem Riemann series theorem Riemann solver Riemann sphere Riemann sum Riemann surface Riemann zeta function Riemann's differential equation Riemann's minimal surface Riemannian circle Riemannian connection on a surface Riemannian geometry Riemann–Hilbert correspondence Riemann–Hilbert problems Riemann–Lebesgue lemma Riemann–Liouville integral Riemann–Roch theorem Riemann–Roch theorem for smooth manifolds Riemann–Siegel formula Riemann–Siegel theta function Riemann–Silberstein vector Riemann–Stieltjes integral Riemann–von Mangoldt formula
Category

Bernhard Riemann