Generalized Riemann hypothesis

The Generalized Riemann hypothesis asserts that all nontrivial zeros of Dirichlet L-function L(\chi,s) for primitive Dirichlet character \chi have real part \frac{1}{2}. The generalized Riemann hypothesis for Dirichlet L-functions was probably formulated for the first time by Adolf Piltz in 1884. It is important to assume primitivity of the character, since for nonprimitive characters, L-functions have infinitely many zeros off this line and do not satisfy the functional equation that is used to distinguish between trivial and nontrivial zeros. Background A Dirichlet character \chi:\mathbb{Z}\rightarrow \mathbb{C} of modulus q is arithmetic 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. For such a character \chi, we define the corresponding Dirichlet L-function by: : L(\chi,s) = \sum_{n=1}^\infty \frac{\chi(n)}{n^s} For every complex number s such that this series is absolutely convergent. By analytic continuation, this function can be extended to a meromorphic function on the complex plane having only a possible pole in s=1, when the character is principal (has only 1 as value for numbers coprime to k). For nonprincipal characters, the series is conditionally convergent for \operatorname{Re}(s)>0 and the analytic continuation is an entire function. We say that the Dirichlet character \chi is inprimitive if it is induced by another Dirichlet character \chi^\star of lesser modulus: : \chi(n) = \begin{cases} \chi^\star(n), & \mathrm{if} \gcd(n,q) = 1 \\ 0, & \mathrm{if} \gcd(n,q) \ne 1 \end{cases} Otherwise we say that the character is primitive. Generally most statements for Dirichlet L-functions are easier to express for versions with primitive characters. Using Euler products of Dirichlet L-functions, we can express the L-function of an imprimitive character \chi as a function of the character \chi^\star that induces it: : L(s,\chi) = L(s,\chi^\star) \prod_{p \,|\, q}\left(1 - \frac{\chi^\star(p)}{p^s} \right) From the factors in this equation we have infinitely many zeros on the line \operatorname{Re}(s)=0. For primitive Dirichlet character, the L-function satisfies a certain functional equation which allows us to define trivial zeros of L(s,\chi) as zeros corresponding to poles of the gamma function in this equation: • If \chi(-1)=1, then all trivial zeros are simple zeros in negative even numbers. If L(s,\chi)\neq \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 called nontrivial zeros. The functional equation guarantees that nontrivial zeros lie in the critical strip 0 and are symmetric with respect to the critical line \operatorname{Re}(s) = \tfrac{1}{2}. The Generalized Riemann Hypothesis says that all nontrivial zeros lie exactly on this line. Consequences of GRH Like the original Riemann hypothesis, the GRH has far-reaching consequences about the distribution of prime numbers: • Taking the trivial character \chi(n)=1 yields the ordinary Riemann hypothesis. • More effective version of Dirichlet's theorem on arithmetic progressions: Let \pi(x,a,d) where a and d are coprime denote the number of prime numbers in arithmetic progression n\cdot d + a which are less than or equal to x. If the generalized Riemann hypothesis is true, then for every : :\pi(x,a,d) = \frac{1}{\varphi(d)} \int_2^x \frac{1}{\ln t}\,dt + O(x^{1/2+\varepsilon})\quad\mbox{ as } \ x\to\infty, :where \varphi is Euler's totient function and O is the Big O notation. This is a considerable strengthening of the prime number theorem. • Every proper subgroup of the multiplicative group (\mathbb Z/n\mathbb Z)^\times has a 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 n less than 3\ln(n)^2. This has many consequences in computational number theory: • In 1976, G. Miller showed that the Miller-Rabin test is guaranteed to run in polynomial time. In 2002, Manindra Agrawal, Neeraj Kayal and Nitin Saxena proved unconditionally that the AKS primality test is guaranteed to run in polynomial time. • The Shanks–Tonelli algorithm is guaranteed to run in polynomial time. • The Ivanyos–Karpinski–Saxena deterministic algorithm for factoring polynomials over finite fields with prime constant-smooth degrees is guaranteed to run in polynomial time. • For every prime p there exists a primitive root mod p (a generator of the multiplicative group of integers modulo p) that is less than O((\ln p)^6). • Estimate of the character sum in the Pólya–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's list 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, see ''''.) • In 1923, Hardy and Littlewood showed that the generalized Riemann hypothesis implies Goldbach weak conjecture for sufficiently large odd numbers. In 1997 Deshouillers, 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 of Harald 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. • In 1934, Chowla showed that the generalized Riemann hypothesis implies that the first prime in the arithmetic progression a mod m is at most Km^2\log(m)^2 for some fixed constant K. • In 1967, Hooley showed that the generalized Riemann hypothesis implies Artin's conjecture on primitive roots. • In 1973, Weinberger showed that the generalized Riemann hypothesis implies that Euler's list of idoneal numbers is complete. • showed that the generalized Riemann hypothesis implies that Ramanujan's integral quadratic form represents all integers that it represents locally, with exactly 18 exceptions. • In 2021, Alexander (Alex) Dunn and Maksym Radziwill proved Patterson's conjecture on cubic Gauss sums, under the assumption of the GRH. == Extended Riemann hypothesis (ERH) ==

Suppose K is a number field with ring of integers O_{K} (this ring is the integral closure of the integers \mathbb Z in K). If I is a nonzero ideal of O_{K}, we denote its norm by N(I). The Dedekind zeta-function of K is then defined by: : \zeta_K(s) = \sum_{I\subseteq O_K} \frac{1}{N(I)^s} for every complex number s 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 \leq \operatorname{Re}(s) \leq 1 and are symmetric with respect to critical line: \operatorname{Re}(s)=\tfrac{1}{2}. The extended Riemann hypothesis asserts that for every number field K each nontrivial zero of \zeta_K has real part \tfrac{1}{2} (and thus lies on the critical line). Consequences of ERH • The ordinary Riemann hypothesis follows from the extended one if one takes the number field to be \mathbb Q, whose ring of integers is: O_{\mathbb Q} = \mathbb Z. • Generalized Riemann hypothesis for Dirichlet L-functions is equivalent to ERH for K being abelian extension of rational numbers, since for abelian extensions \zeta_K is finite product of some Dirichlet L-functions depending on K. Conversely, all L-functions for character modulo n appears in product for K=\mathbb{Q}(\zeta_n), where \zeta_n is n-th primitive root of unity. • For general extensions, similar role to Dirichlet L-functions is played by Artin L-functions. Then, ERH is equivalent to Riemann Hypothesis for Artin L-functions. • The ERH implies an effective version of the Chebotarev density theorem: if L/K is a finite Galois extension with Galois group G, and C a union of conjugacy classes of G, the number of unramified primes of K of norm below x with Frobenius conjugacy class in C is : \frac\Bigl(\operatorname{Li}(x)+O\bigl(\sqrt x(n\log x+\log|\Delta|)\bigr)\Bigr), : where the constant implied in the big-O notation is absolute, n is the degree of L over Q, and Δ its discriminant. • showed that ERH implies that any number field with class number 1 is either Euclidean or an imaginary quadratic number field of discriminant −19, −43, −67, or −163. • discussed how the ERH can be used to give sharper estimates for discriminants and class numbers of number fields. == Generalized Riemann hypothesis for Selberg class ==

Selberg class is defined the following way: We say that Dirichlet series F(s)=\sum_{n=1}^{\infty}\frac{a_n}{n^s} is in Selberg class if it satisfies following properties: • Analyticity: 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 a_n \ll_\varepsilon n^\varepsilon for any ε > 0; • Functional equation: there is a gamma factor of the form : \gamma(s)=Q^s\prod_{i=1}^k \Gamma (\omega_is+\mu_i) where Q is real and positive, \Gamma the gamma 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: : \Phi(s) = \gamma(s) F(s)\, satisfies: : \Phi(s)=\alpha\,\overline{\Phi(1-\overline{s})}; • Euler product: For , F(s) can be written as a product over primes: : F(s)=\prod_p F_p(s) with : F_p(s)=\exp\left(\sum_{n=1}^\infty\frac{b_{p^n}}{p^{ns}}\right) and, for some \theta , : b_{p^n}=O(p^{n\theta}). From analyticity follows that poles of gamma factor in \operatorname{Re}(s) 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 0 and are symmetric with respect to critical line \operatorname{Re}(s)=\tfrac{1}{2}. Generalized Riemann hypothesis for Selberg class states that all nontrivial zeros of function F belonging to Selberg class have real part \tfrac{1}{2} and then lie on critical line. Selberg class along with proposition of Riemann hypothesis for it was firs introduced in . Instead of considering specific functions, Selberg approach was to give axiomatic definition consisting of properties characterizing most of objects called L-functions or zeta functions and expected to satisfy counterparts or generalizations of Riemann hypothesis. Consequences • Artin L-functions and Dedekind zeta functions belong to Selberg class, then Riemann Hypothesis for Selberg class implies extended Riemann hypothesis. • Nontrivial zeros for much more general L-functions than Dedekind zeta functions lie on critical lines. One example can be Ramanujan L-function related to modular form called Dedekind eta function. Despite Ramanujan L-function itself don't belong to Selberg class and its critical line is \operatorname{Re}(s)=6, function obtained by translation of \tfrac{11}{2} is in Selberg class. == See also ==