TheRiemann hypothesis is one of the most importantconjectures inmathematics. It is a statement about the zeros of theRiemann zeta function. Various geometrical and arithmetical objects can be described by so-called globalL-functions, which are formally similar to the Riemann zeta-function. One can then ask the same question about the zeros of theseL-functions, yielding various generalizations of the Riemann hypothesis. Many mathematicians believe these generalizations of the Riemann hypothesis to be true. The only cases of these conjectures which have been proven occur in thealgebraic function field case (not the number field case).

GlobalL-functions can be associated toelliptic curves,number fields (in which case they are calledDedekind zeta-functions),Maass forms, andDirichlet characters (in which case they are calledDirichletL-functions). When the Riemann hypothesis is formulated for Dedekind zeta-functions, it is known as theextended Riemann hypothesis (ERH) and when it is formulated for DirichletL-functions, it is known as thegeneralised Riemann hypothesis (GRH). Another approach to generalization of Riemann hypothesis was given byAtle Selberg and his introduction of class of function satisfying certain properties rather than specific functions, nowadays known as Selberg class. These three statements will be discussed in more detail below. (Many mathematicians use the labelgeneralized Riemann hypothesis to cover the extension of the Riemann hypothesis to all globalL-functions, not only the special case of DirichletL-functions.)

Generalized Riemann hypothesis asserts that all nontrivial zeros of DirichletL-function$L(\chi ,s)$ for primitive Dirichlet character${\displaystyle \chi }$ have real part$\frac{1}{2}$.

The generalized Riemann hypothesis for DirichletL-functions was probably formulated for the first time byAdolf Piltz in 1884.^[1]It is important to assume primitivity of character since for nonprimitive charactersL-functions have infinitely many zeros off this line and don't satisfy functional equation that is used to distinguish between trivial and nontrivial zeros.

ADirichlet character$\chi :\mathbb{Z}\to \mathbb{C}$ of modulusq isarithmetic function that is:

• completely multiplicative:$\chi (a\cdot b)=\chi (a)\cdot \chi (b)$
• periodic:$\chi (n+q)=\chi (n)$
• $\chi (n)=0$ if and only if$gcd(n,q)>1$.

If such character$\chi$, we define the corresponding DirichletL-function by:

${\displaystyle L(\chi ,s)=\sum _{n=1}^{\mathrm{\infty }}\frac{\chi (n)}{n^s}}$

For everycomplex numbers such thatRes > 1 this series is absolutely convergent. Byanalytic continuation, this function can be extended tomeromorphic function on complex plane having only possible pole in$s=1$, when character is principal (have only 1 as value for numbers coprime tok). For nonprincipal character, series is conditionally convergent for$\mathrm{Re}(s)>0$ and analytic continuation isentire function.

We say that Dirichlet character$\chi$ isinprimitive if it is induced by another Dirichlet${\chi }^{\star }$ character of lesser modulus:

Otherwise we say that character isprimitive. Generally most statements for Dirichlet L-functions are easier to express for versions with primitive characters. Using Euler product of Dirichlet L-functions we can express L-function of imprimitive character$\chi$ by function of character${\chi }^{\star }$ that induces it:

${\displaystyle L(s,\chi )=L(s,{\chi }^{\star })\prod _{p|q}\left(1-\frac{{\chi }^{\star }(p)}{p^s}\right)}$

From factors in this equation we have infinitely many zeros on line:$\mathrm{Re}(s)=0$.For primitive Dirichlet character L-function satisfies certainfunctional equation which allows us to definetrivial zeros of$L(s,\chi )$ as zeros corresponding to poles of gamma function in this equation:

• If$\chi (-1)=1$, then all trivial zeros are simple zeros in negative even numbers. If$L(s,\chi )\ne \zeta (s)$ it also includes 0.
• If$\chi (-1)=-1$ then all trivial zeros are simple zeros in negative odd numbers.

Any other zeros are callednontrivial zeros. Functional equation guarantees that nontrivial zeros lies in critical strip: and are symmetric with respect to critical line$\mathrm{Re}(s)=\frac{1}{2}$. Generalized Riemann Hypothesis says that all nontrivial zeros lies exactly on this line.

Like the original Riemann hypothesis, GRH has far-reaching consequences about the distribution ofprime numbers:

• Taking trivial character$\chi (n)=1$ yields the ordinary Riemann hypothesis.
• More effective version ofDirichlet's theorem on arithmetic progressions: Let$\pi (x,a,d)$ wherea andd are coprime denote the number of prime numbers in arithmetic progression$n\cdot d+a$ which are less than or equal tox. If the generalized Riemann hypothesis is true, then for everyε > 0:

${\displaystyle \pi (x,a,d)=\frac{1}{\phi (d)}{\int }_2^x\frac{1}{\ln t}dt+O(x^{1/2+\epsilon })\text{ as }x\to \mathrm{\infty },}$

where${\displaystyle \phi }$ isEuler's totient function and${\displaystyle O}$ is theBig O notation. This is a considerable strengthening of theprime number theorem.

• Every proper subgroup of the multiplicative group$(\mathbb{Z}/n\mathbb{Z}{)}^{\times }$ has set of generators less than$2\ln (n{)}^2$. In other words, every subgroup of multiplicative group omits a number less than$2\ln (n{)}^2$, as well as a number coprime to${\displaystyle n}$ less than$3\ln (n{)}^2$.^[2] This has many consequences incomputational number theory:
  • In 1976, G. Miller showed thatMiller-Rabin test is guaranteed to run in polynomial time. In 2002, Manindra Agrawal, Neeraj Kayal and Nitin Saxena proved unconditionally thatAKS primality test is guaranteed to run in polynomial time.
  • TheShanks–Tonelli algorithm is guaranteed to run in polynomial time.
  • The Ivanyos–Karpinski–Saxena deterministic algorithm^[3] for factoring polynomials over finite fields with prime constant-smooth degrees is guaranteed to run in polynomial time.
• For every primep there exists aprimitive root modp (a generator of the multiplicative group of integers modulop) that is less than${\displaystyle O((\ln p{)}^6).}$^[4]
• Estimate of the character sum in thePólya–Vinogradov inequality can be improved to$O\left(\sqrt{q}\log \log q\right)$,q being the modulus of the character.
• 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$${\displaystyle \underset{x\to 1^{-}}{lim}\sum _{p>2}(-1{)}^{(p+1)/2}{x}^p=+\mathrm{\infty },}$$ 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 impliesGoldbach weak conjecture for sufficiently large odd numbers. In 1997Deshouillers, Effinger,te Riele, and Zinoviev showed that actually 5 is sufficiently large, so GRH implies weak Goldbach conjecture. In 1937 Vinogradov gave an unconditional proof for sufficiently large odd numbers. The yet to be verified proof ofHarald Helfgott improved Vinogradov's method by verifying GRH for several thousand small characters up to a certain imaginary part to prove the conjecture for all integers above 10²⁹, integers below which have already been verified by calculation.^[5]
• In 1934, Chowla showed that the generalized Riemann hypothesis implies that the first prime in the arithmetic progressiona modm is at most$K{m}^2\log (m{)}^2$ 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.
• 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.^[6][7]

Suppose$K$ is anumber field withring of integers$O_K$ (this ring is theintegral closure of theintegers${\displaystyle \mathbb{Z}}$ inK). If$I$ is a nonzeroideal of$O_K$, we denote itsnorm by$N(I)$. TheDedekind zeta-function ofK is then defined by:

${\displaystyle {\zeta }_K(s)=\sum _{I\subseteq O_K}\frac{1}{N(I{)}^s}}$

for every complex numbers with real part > 1. The sum extends over all non-zero ideals$I$ of$O_K$. That function can be extended by analytic continuation to the meromorphic function on complex plane with only possible pole at$s=1$ and satisfies a functional equation that gives exact location of trivial zeroes and guarantees that nontrivial zeros lie inside critical strip$0\le \mathrm{Re}(s)\le 1$ and are symmetric with respect to critical line:$\mathrm{Re}(s)=\frac{1}{2}$.

Theextended Riemann hypothesis asserts that for every number fieldK each nontrivial zero of${\zeta }_K$ has real part$\frac{1}{2}$ (and thus lies on the critical line).

• The ordinary Riemann hypothesis follows from the extended one if one takes the number field to be${\displaystyle \mathbb{Q}}$, whose ring of integers is:${\displaystyle O_{\mathbb{Q}}=\mathbb{Z}}$.
• Generalized Riemann hypothesis for Dirichlet L-functions is equivalent to ERH forK beingabelian extension of rational numbers, since for abelian extensions${\zeta }_K$ is finite product of some Dirichlet L-functions depending onK. Conversely, all L-functions for character modulon appears in product for$K=\mathbb{Q}({\zeta }_n)$, where${\zeta }_n$ isn-th primitive root of unity.
• For general extensions, similar role to Dirichlet L-functions is played byArtin L-functions. Then, ERH is equivalent to Riemann Hypothesis for Artin L-functions.
• The ERH implies an effective version^[8] of theChebotarev density theorem: ifL/K is a finite Galois extension with Galois groupG, andC a union of conjugacy classes ofG, the number ofunramified primes ofK of norm belowx with Frobenius conjugacy class inC is

${\displaystyle \frac{|C|}{|G|}(\mathrm{Li}(x)+O(\sqrt{x}(n\log x+\log |\mathrm{\Delta }|))),}$

where the constant implied in the big-O notation is absolute,n is the degree ofL overQ, and Δ its discriminant.

• Weinberger (1973) showed that ERH implies that any number field with class number 1 is eitherEuclidean or an imaginary quadratic number field ofdiscriminant −19, −43, −67, or −163.
• Odlyzko (1990) discussed how the ERH can be used to give sharper estimates for discriminants and class numbers of number fields.

Selberg class is defined the following way:

We say that Dirichlet series$F(s)=\sum _{n=1}^{\mathrm{\infty }}\frac{a_n}{n^s}$ is in Selberg class if it satisfies following properties:

• Analyticity:${\displaystyle F(s)}$ has a meromorphic continuation to the entire complex plane, with the only possible pole (if any) in$s=1$.
• Ramanujan conjecture:a₁ = 1 and${\displaystyle a_n{\ll }_{\epsilon }{n}^{\epsilon }}$ for any ε > 0;
• Functional equation: there is a gamma factor of the form

${\displaystyle \gamma (s)=Q^s\prod _{i=1}^k\mathrm{\Gamma }({\omega }_{i}s+{\mu }_i)}$

where$Q$ is real and positive,$\mathrm{\Gamma }$ thegamma function, the${\omega }_i$ real and positive, and the${\mu }_i$ complex with non-negative real part, as well as a so-called root number:$\alpha \in \mathbb{C},|\alpha |=1$, such that the function:

${\displaystyle \mathrm{\Phi }(s)=\gamma (s)F(s)}$

satisfies:

${\displaystyle \mathrm{\Phi }(s)=\alpha \overline{\mathrm{\Phi }(1-\overline{s})};}$

• Euler product: ForRe(s) > 1,F(s) can be written as a product over primes:

${\displaystyle F(s)=\prod _p{F}_p(s)}$

with

${\displaystyle F_p(s)=\exp \left(\sum _{n=1}^{\mathrm{\infty }}\frac{b_{p^n}}{p^{ns}}\right)}$

and, for some,

${\displaystyle b_{p^n}=O(p^{n\theta }).}$

From analyticity follows that poles of gamma factor in must be cancelled by zeros of$F(s)$, that zeros are called trivial zeros. Functional equation guarantees that all nontrivial zeros lie in critical strip and are symmetric with respect to critical line$\mathrm{Re}(s)=\frac{1}{2}$.

Generalized Riemann hypothesis for Selberg class states that all nontrivial zeros of function$F$ belonging to Selberg class have real part$\frac{1}{2}$ and then lie on critical line.

Selberg class along with proposition of Riemann hypothesis for it was firs introduced in (Selberg 1992). Instead of considering specific functions, Selberg approach was to give axiomatic definition consisting of properties characterizing most of objects calledL-functions orzeta functions and expected to satisfy counterparts or generalizations of Riemann hypothesis.

• ArtinL-functions and Dedekind zeta functions belong to Selberg class, then Riemann Hypothesis for Selberg class implies extended Riemann hypothesis.
• Nontrivial zeros for much more generalL-functions than Dedekind zeta functions lie on critical lines. One example can beRamanujanL-function related to modular form calledDedekind eta function. Despite Ramanujan L-function itself don't belong to Selberg class and its critical line is${\displaystyle \mathrm{Re}(s)=6}$, function obtained by translation of${\displaystyle \frac{11}{2}}$ is in Selberg class.

• Artin's conjecture
• Artin L-function
• Dirichlet L-function
• Dedekind zeta function
• Selberg class
• Grand Riemann hypothesis

• Ono, Ken;Soundararajan, K. (1997), "Ramanujan's ternary quadratic form",Inventiones Mathematicae,130 (3):415–454,Bibcode:1997InMat.130..415O,doi:10.1007/s002220050191,S2CID 122314044
• Odlyzko, A. M. (1990),"Bounds for discriminants and related estimates for class numbers, regulators and zeros of zeta functions: a survey of recent results",Séminaire de Théorie des Nombres de Bordeaux, Série 2,2 (1):119–141,doi:10.5802/jtnb.22 (inactive 17 December 2025),MR 1061762{{citation}}: CS1 maint: DOI inactive as of December 2025 (link)
• Weinberger, Peter J. (1973), "On Euclidean rings of algebraic integers",Analytic number theory ( St. Louis Univ., 1972), Proc. Sympos. Pure Math., vol. 24, Providence, R.I.: Amer. Math. Soc., pp. 321–332,MR 0337902
• Selberg, Atle (1992), "Old and new conjectures and results about a class of Dirichlet series",Proceedings of the Amalfi Conference on Analytic Number Theory (Maiori, 1989), Salerno: Univ. Salerno, pp. 367–385,MR 1220477,Zbl 0787.11037 Reprinted in Collected Papers, vol2, Springer-Verlag, Berlin (1991)

• "Riemann hypothesis, generalized",Encyclopedia of Mathematics,EMS Press, 2001 [1994]

• Zeta and L-functions
• Algebraic geometry
• Conjectures
• Unsolved problems in mathematics
• Bernhard Riemann

• Articles with short description
• Short description is different from Wikidata
• CS1 maint: DOI inactive as of December 2025