The values of the Riemann zeta function at even positive integers were computed by Euler. The first of them,ζ(2), provides a solution to theBasel problem. In 1979,Roger Apéry proved the irrationality ofζ(3), and got the number named after him. The values at negative integer points, also found by Euler, arerational numbers and play an important role in the theory ofmodular forms. Many generalizations of the Riemann zeta function, such asDirichlet series,DirichletL-functions andL-functions, are known.
Bernhard Riemann's articleOn the number of primes below a given magnitude
The Riemann zeta functionζ(s) is a function of a complex variables =σ +it, whereσ andt are real numbers. (The notations,σ, andt is used traditionally in the study of the zeta function, following Riemann.) WhenRe(s) =σ > 1, the function can be written as a converging summation or as an integral:
where
is thegamma function. The Riemann zeta function is defined for other complex values viaanalytic continuation of the function defined forσ > 1.
Leonhard Euler considered the above series in 1740 for positive integer values ofs, and laterChebyshev extended the definition toRe(s) > 1.[4]
The above series is a prototypicalDirichlet series thatconverges absolutely to ananalytic function fors such thatσ > 1 anddiverges for all other values ofs. Riemann showed that the function defined by the series on the half-plane of convergence can be continued analytically to all complex valuess ≠ 1. Fors = 1, the series is theharmonic series which diverges to+∞, andThus the Riemann zeta function is ameromorphic function on the whole complex plane, which isholomorphic everywhere except for asimple pole ats = 1 withresidue1.
where, by definition, the left hand side isζ(s) and theinfinite product on the right hand side extends over all prime numbersp (such expressions are calledEuler products):
Both sides of the Euler product formula converge forRe(s) > 1. Theproof of Euler's identity uses only the formula for thegeometric series and thefundamental theorem of arithmetic. Since theharmonic series, obtained whens = 1, diverges, Euler's formula (which becomesΠpp/p − 1) implies that there areinfinitely many primes.[5] Since the logarithm ofp/(p − 1) is approximately1/p, the formula can also be used to prove the stronger result that the sum of the reciprocals of the primes is infinite. On the other hand, combining that with thesieve of Eratosthenes shows that the density of the set of primes within the set of positive integers is zero.
The Euler product formula can be used to calculate theasymptotic probability thats randomly selected integers within a bound are set-wisecoprime. Intuitively, the probability that any single number is divisible by a prime (or any integer)p is1/p. Hence the probability thats numbers are all divisible by this prime is1/ps, and the probability that at least one of them isnot is1 − 1/ps. Now, for distinct primes, these divisibility events are mutually independent because the candidate divisors are coprime (a number is divisible by coprime divisorsn andmif and only if it is divisible by nm, an event which occurs with probability 1/(nm)). Thus the asymptotic probability thats numbers are coprime is given by a product over all primes,[6]
This zeta function satisfies thefunctional equationwhereΓ(s) is thegamma function. This is an equality of meromorphic functions valid on the wholecomplex plane. The equation relates values of the Riemann zeta function at the pointss and1 −s, in particular relating even positive integers with odd negative integers. Owing to the zeros of the sine function, the functional equation implies thatζ(s) has a simple zero at each even negative integers = −2n, known as thetrivial zeros ofζ(s). Whens is an even positive integer, the product on the right is non-zero becauseΓ(1 −s) has a simplepole, which cancels the simple zero of the sine factor. Whens is0, the zero of the sine factor is cancelled by the simple pole ofζ(1).
Proof of Riemann's functional equation
A proof of the functional equation proceeds as follows:We observe that ifs > 0, then
As a result, ifs > 1 thenwith the inversion of the limiting processes justified by absolute convergence (hence the stricter requirement on).
For convenience, letwhich is a special case of thetheta function.
Sowhich is convergent for alls, becauseψ(x) → 0 more quickly than any power ofx forx > 1, so the integral converges. As the RHS remains the same ifs is replaced by1 −s,which is the functional equation attributed toBernhard Riemann.[8]
Using integration by parts again with a factorization ofx3/2,
As,
Remove a factor ofx−1/4 to make the exponents in the remainder opposites.
Using thehyperbolic functions, namelycos(x) = cosh(ix), and lettings = 1/2 +it givesand by separating the integral and using thepower series forcos,which led Riemann to his famous hypothesis.
Zeros, the critical line, and the Riemann hypothesis
The Riemann zeta function has no zeros to the right ofσ = 1 or (apart from the trivial zeros) to the left ofσ = 0 (nor can the zeros lie too close to those lines). Furthermore, the non-trivial zeros are symmetric about the real axis and the lineσ = 1/2 and, according to theRiemann hypothesis, they all lie on the lineσ = 1/2.This image shows a plot of the Riemann zeta function along the critical line for real values oft running from 0 to 34. The first five zeros in the critical strip are clearly visible as the place where the spirals pass through the origin.The real part (red) and imaginary part (blue) of the Riemann zeta function along the critical lineRe(s) = 1/2. The first non-trivial zeros can be seen atIm(s) =±14.135,±21.022 and±25.011.
The functional equation shows that the Riemann zeta function has zeros at−2, −4, .... These are called thetrivial zeros. They are trivial in the sense that their existence is relatively easy to prove, for example, fromsin(πs/2) being0 in the functional equation. The non-trivial zeros have captured far more attention because their distribution not only is far less understood but, more importantly, their study yields important results concerning prime numbers and related objects in number theory. It is known that any non-trivial zero lies in the open strip{s ∈ | 0 < Re(s) < 1}, which is called thecritical strip. The set{s ∈ | Re(s) = 1/2} is called thecritical line. TheRiemann hypothesis, considered one of the greatest unsolved problems in mathematics, asserts that all non-trivial zeros are on the critical line. In 1989, Conrey proved that more than 40% of the non-trivial zeros of the Riemann zeta function are on the critical line.[9] This has since been improved to 41.7%.[10]
For the Riemann zeta function on the critical line, seeZ-function.
LetN(T) be the number of zeros ofζ(s) in the critical strip0 < Re(s) < 1, whose imaginary parts are in the interval0 < Im(s) <T.Timothy Trudgian proved that, ifT >e, then[13]
In 1914,G. H. Hardy proved thatζ(1/2 +it) has infinitely many real zeros.[14][15]
Hardy andJ. E. Littlewood formulated two conjectures on the density and distance between the zeros ofζ(1/2 +it) on intervals of large positive real numbers. In the following,N(T) is the total number of real zeros andN0(T) the total number of zeros of odd order of the functionζ(1/2 +it) lying in the interval(0,T].
For anyε > 0, there exists aT0(ε) > 0 such that when
the interval(T,T +H] contains a zero of odd order.
For anyε > 0, there exists aT0(ε) > 0 andcε > 0 such that the inequality
holds when
These two conjectures opened up new directions in the investigation of the Riemann zeta function.
The location of the Riemann zeta function's zeros is of great importance in number theory. Theprime number theorem is equivalent to the fact that there are no zeros of the zeta function on the lineRe(s) = 1.[16] It is also known that zeros do not exist in certain regions slightly to the left of the lineRe(s) = 1, known as zero-free regions. For instance, Korobov[17] and Vinogradov[18] independently showed via theVinogradov's mean-value theorem that for sufficiently large|t|,ζ(σ +it) ≠ 0 for
for anyε > 0 and a numberc > 0 depending onε. Asymptotically, this is the largest known zero-free region for the zeta function.
Explicit zero-free regions are also known. Platt and Trudgian[19]verified computationally thatζ(σ +it) ≠ 0 ifσ ≠ 1/2 and|t| ≤ 3⋅1012. Mossinghoff, Trudgian and Yang proved[20] that zeta has no zeros in the region
for|t| ≥ 2, which is the largest known zero-free region in the critical strip for3⋅1012 < |t| < exp(64.1) ≈ 7⋅1027 (for previous results see[21]).Yang[22] showed thatζ(σ +it) ≠ 0 if
and
which is the largest known zero-free region forexp(170.2) < |t| < exp(4.8⋅105).Bellotti proved[23] (building on the work of Ford[24]) the zero-free region
and.
This is the largest known zero-free region for fixed|t| ≥ exp(4.8⋅105). Bellotti also showed that for sufficiently large|t|, the following better result is known:ζ(σ +it) ≠ 0 for
The strongest result of this kind one can hope for is the truth of the Riemann hypothesis, which would have many profoundconsequences in the theory of numbers.
It is known that there are infinitely many zeros on the critical line.Littlewood showed that if the sequence (γn) contains the imaginary parts of all zeros in theupper half-plane in ascending order, then
Thecritical line theorem asserts that a positive proportion of the nontrivial zeros lies on the critical line. (The Riemann hypothesis would imply that this proportion is1.)
In the critical strip, the zero with smallest non-negative imaginary part is1/2 + 14.13472514...i (OEIS: A058303). The fact that, for all complexs ≠ 1,
implies that the zeros of the Riemann zeta function are symmetric about the real axis. Combining this symmetry with the functional equation, furthermore, one sees that the non-trivial zeros are symmetric about the critical lineRe(s) = 1/2.
It is also known that no zeros lie on the line with real part1.
A large class of modified zeta functions exists that share the same non-trivial zeros as the Riemann zeta function, where modification means replacing the prime numbers in the Euler product by real numbers, which was shown in aresult by Grosswald and Schnitzer.
For any positive even integer2n,whereB2n is the(2n)thBernoulli number.For odd positive integers, no such simple expression is known, although these values are thought to be related to the algebraicK-theory of the integers; seeSpecial values ofL-functions.
For nonpositive integers, one hasforn ≥ 0 (using the convention thatB1 = 1/2).In particular,ζ vanishes at the negative even integers becauseBm = 0 for all oddm other than 1. These are the so-called "trivial zeros" of the zeta function.
The demonstration of the particular valueis known as theBasel problem. The reciprocal of this sum answers the question: 'What is the probability that two numbers selected from a uniform distribution from1 ton] arecoprime asn → ∞?'[29]The valueisApéry's constant.
for every complex numbers with real part greater than1. There are a number of similar relations involving various well-knownmultiplicative functions; these are given in the article on theDirichlet series.
The Riemann hypothesis is equivalent to the claim that this expression is valid when the real part ofs is greater than1/2.
The critical strip of the Riemann zeta function has the remarkable property ofuniversality. Thiszeta function universality states that there exists some location on the critical strip that approximates anyholomorphic function arbitrarily well. Since holomorphic functions are very general, this property is quite remarkable. The first proof of universality was provided bySergei Mikhailovitch Voronin in 1975.[30] More recent work has includedeffective versions of Voronin's theorem[31] andextending it toDirichletL-functions.[32][33]
Estimates of the maximum of the modulus of the zeta function
Let the functionsF(T;H) andG(s0; Δ) be defined by the equalities
HereT is a sufficiently large positive number,0 <H ≪ log logT,s0 =σ0 +iT,1/2 ≤σ0 ≤ 1,0 < Δ < 1/3. Estimating the valuesF andG from below shows, how large (in modulus) valuesζ(s) can take on short intervals of the critical line or in small neighborhoods of points lying in the critical strip0 ≤ Re(s) ≤ 1.
The caseH ≫ log logT was studied byKanakanahalli Ramachandra; the caseΔ >c, wherec is a sufficiently large constant, is trivial.
Anatolii Karatsuba proved,[34][35] in particular, that if the valuesH andΔ exceed certain sufficiently small constants, then the estimates
hold, wherec1 andc2 are certain absolute constants.
is called theargument of the Riemann zeta function. Hereargζ(1/2 +it) is the increment of an arbitrary continuous branch ofargζ(s) along the broken line joining the points2,2 +it and1/2 +it.
There are some theorems on properties of the functionS(t). Among those results[36][37] are themean value theorems forS(t) and its first integral
on intervals of the real line, and also the theorem claiming that every interval(T,T +H] for
contains at least
points where the functionS(t) changes sign. Earlier similar results were obtained byAtle Selberg for the case
An extension of the area of convergence can be obtained by rearranging the original series.[38] The series
converges forRe(s) > 0, while
converge even forRe(s) > −1. In this way, the area of convergence can be extended toRe(s) > −k for any negative integer−k.
The recurrence connection is clearly visible from the expression valid forRe(s) > −2 enabling further expansion by integration by parts.
This recurrence leads to this other series development that uses therising factorial and is valid for the entire complex plane[38]
This can be used recursively to extend the Dirichlet series definition to all complex numbers.
The Riemann zeta function also appears in a form similar to the Mellin transform in an integral over theGauss–Kuzmin–Wirsing operator acting onxs−1; that context gives rise to a series expansion in terms of thefalling factorial.[39]
in the region where the integral is defined. There are various expressions for the zeta function as Mellin transform-like integrals. If the real part ofs is greater than one, we have
A similar Mellin transform involves the Riemann functionJ(x), which counts prime powerspn with a weight of1/n, so that
Now
These expressions can be used to prove the prime number theorem by means of the inverse Mellin transform. Riemann'sprime-counting function is easier to work with, andπ(x) can be recovered from it byMöbius inversion.
However, this integral only converges if the real part ofs is greater than1, but it can be regularized. This gives the following expression for the zeta function, which is well defined for alls except0 and1:
The Riemann zeta function ismeromorphic with a singlepole of order one ats = 1. It can therefore be expanded as aLaurent series abouts = 1; the series development is then[43]
This form clearly displays the simple pole ats = 1, the trivial zeros at−2, −4,... due to the gamma function term in the denominator, and the non-trivial zeros ats =ρ. (To ensure convergence in the latter formula, the product should be taken over "matching pairs" of zeros, i.e. the factors for a pair of zeros of the formρ and1 −ρ should be combined.)
A globally convergent series for the zeta function, valid for all complex numberss excepts = 1 +2πi/ln 2n for some integern, was conjectured byKonrad Knopp in 1926[44] and proven byHelmut Hasse in 1930[45] (cf.Euler summation):
The series appeared in an appendix to Hasse's paper, and was published for the second time by Jonathan Sondow in 1994.[46]
Hasse also proved the globally converging series
in the same publication.[45] Research by Iaroslav Blagouchine[47][44]has found that a similar, equivalent series was published byJoseph Ser in 1926.[48]
In 1997 K. Maślanka gave another globally convergent (excepts = 1) series for the Riemann zeta function:
where real coefficients are given by:
HereBn are the Bernoulli numbers and(x)k denotes the Pochhammer symbol.[49][50]
Note that this representation of the zeta function is essentially an interpolation with nodes, where the nodes are pointss = 2, 4, 6, ..., i.e. exactly those where the zeta values are precisely known, as Euler showed. An elegant and very short proof of this representation of the zeta function, based onCarlson's theorem, was presented by Philippe Flajolet in 2006.[51]
The asymptotic behavior of the coefficients is rather curious: for growing values, we observe regular oscillations with a nearly exponentially decreasing amplitude and slowly decreasing frequency (roughly as). Using the saddle point method, we can show that
On the basis of this representation, in 2003 Luis Báez-Duarte provided a new criterion for the Riemann hypothesis.[53][54][55] Namely, if we define the coefficientsck as
There are a number of relatedzeta functions that can be considered to be generalizations of the Riemann zeta function. These include theHurwitz zeta function
One can analytically continue these functions to then-dimensional complex space. The special values taken by these functions at positive integer arguments are calledmultiple zeta values by number theorists and have been connected to many different branches in mathematics and physics.
^Devlin, Keith (2002).The Millennium Problems: The seven greatest unsolved mathematical puzzles of our time. New York: Barnes & Noble. pp. 43–47.ISBN978-0-7607-8659-8.
^Sandifer, Charles Edward (2007).How Euler Did It. Mathematical Association of America. p. 193.ISBN978-0-88385-563-8.
^Trudgian, Timothy S. (2014). "An improved upper bound for the argument of the Riemann zeta function on the critical line II".J. Number Theory.134:280–292.arXiv:1208.5846.doi:10.1016/j.jnt.2013.07.017.
^Hardy, G.H. (1914). "Sur les zeros de la fonction ζ(s)".Comptes rendus de l'Académie des Sciences.158.French Academy of Sciences:1012–1014.
^Korobov, Nikolai Mikhailovich (1958). "Estimates of trigonometric sums and their applications".Usp. Mat. Nauk.13 (4):185–192.
^Vinogradov, I.M. (1958). "Eine neue Abschätzung der Funktionζ(1 +it)".Russian. Izv. Akad. Nauk SSSR, Ser. Mat.22:161–164.
^Platt, David; Trudgian, Timothy S. (2021). "The Riemann hypothesis is true up to 3⋅1012".Bulletin of the London Mathematical Society.53 (3):792–797.arXiv:2004.09765.doi:10.1112/blms.12460.
^Mossinghoff, Michael J.; Trudgian, Timothy S.; Yang, Andrew (2024). "Explicit zero-free regions for the Riemann zeta-function".Res. Number Theory.10 11.arXiv:2212.06867.doi:10.1007/s40993-023-00498-y.
^Mossinghoff, Michael J.; Trudgian, Timothy S. (2015). "Nonnegative trigonometric polynomials and a zero-free region for the Riemann zeta-function".J. Number Theory.157:329–349.arXiv:1410.3926.doi:10.1016/J.JNT.2015.05.010.S2CID117968965.
^Bellotti, Chiara (2024). "Explicit bounds for the Riemann zeta function and a new zero-free region".J. Math. Anal. Appl.536 (2) 128249.arXiv:2306.10680.doi:10.1016/j.jmaa.2024.128249.
^Kainz, A. J.; Titulaer, U. M. (1992). "An accurate two-stream moment method for kinetic boundary layer problems of linear kinetic equations".J. Phys. A: Math. Gen.25 (7):1855–1874.Bibcode:1992JPhA...25.1855K.doi:10.1088/0305-4470/25/7/026.
^Further digits and references for this constant are available atOEIS: A059750.
^Voronin, S. M. (1975). "Theorem on the Universality of the Riemann Zeta Function".Izv. Akad. Nauk SSSR, Ser. Matem.39:475–486. Reprinted inMath. USSR Izv. (1975)9: 443–445.
^Maślanka, Krzysztof; Koleżyński, Andrzej (2022). "The High Precision Numerical Calculation of Stieltjes Constants. Simple and Fast Algorithm".Computational Methods in Science and Technology.28 (2):47–59.arXiv:2210.04609.doi:10.12921/cmst.2022.0000014.S2CID252780397.
^William A. Sethares (2005).Tuning, Timbre, Spectrum, Scale (2nd ed.). Springer-Verlag London. p. 74.... there are many different ways to evaluate the goodness, reasonableness, fitness, or quality of a scale ... Under some measures, 12-tet is the winner, under others 19-tet appears best, 53-tet often appears among the victors ...
^Most of the formulas in this section are from § 4 of J. M. Borwein et al. (2000)
Hasse, Helmut (1930). "Ein Summierungsverfahren für die Riemannscheζ-Reihe" [A summation method for the Riemannζ series].Math. Z. (in German).32:458–464.doi:10.1007/BF01194645.MR1545177.S2CID120392534. (Globally convergent series expression.)
Raoh, Guo (1996). "The distribution of the logarithmic derivative of the Riemann zeta function".Proceedings of the London Mathematical Society. S3–72:1–27.doi:10.1112/plms/s3-72.1.1.
