Generalized Riemann hypothesis
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The Riemann hypothesis is one of the most important conjectures in mathematics. It is a statement about the zeros of the Riemann zeta function. Various geometrical and arithmetical objects can be described by so-called global L-functions, which are formally similar to the Riemann zeta function. One can then ask the same question about the zeros of these L-functions, yielding various generalizations of the Riemann hypothesis. None of these conjectures have been proven or disproven, but many mathematicians believe them to be true.
Global L-functions can be associated to elliptic curves, number fields (in which case they are called Dedekind zeta functions), Maass waveforms, and Dirichlet characters (in which case they are called Dirichlet L-functions). When the Riemann hypothesis is formulated for Dedekind zeta functions, it is known as the extended Riemann hypothesis and when it is formulated for Dirichlet L-functions, it is known as the generalized Riemann hypothesis. These two statements will be discussed in more detail below.
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Generalized Riemann hypothesis (GRH)
The generalized Riemann hypothesis was probably formulated for the first time by Piltz in 1884. Like the original Riemann hypothesis, it has far reaching consequences about the distribution of prime numbers.
The formal statement of the hypothesis follows. A Dirichlet character is a completely multiplicative arithmetic function χ such that there exists a positive integer k with χ(n + k) = χ(n) for all n and χ(n) = 0 whenever gcd(n, k) > 1. If such a character is given, we define the corresponding Dirichlet L-function by
- <math>
L(\chi,s) = \sum_{n=1}^\infty \frac{\chi(n)}{n^s} <math>
for every complex number s with real part > 1. By analytic continuation, this function can be extended to a meromorphic function defined on the whole complex plane. The generalized Riemann hypothesis asserts that for every Dirichlet character χ and every complex number s with L(χ,s) = 0: if the real part of s is between 0 and 1, then it is actually 1/2.
The case χ(n) = 1 for all n yields the ordinary Riemann hypothesis.
Consequences of GRH
An arithmetic progression in the natural numbers is a set of numbers of the form a, a+d, a+2d, a+3d, ... where a and d are natural numbers and d is non-zero. Dirichlet's theorem states that if a and d are coprime, then such an arithmetic progression contains infinitely many prime numbers. Let π(x,a,d) denote the number of prime numbers in this progression which are less than or equal to x. If the generalized Riemann hypothesis is true, then for every coprime a and d and for every ε > 0
- <math>
\pi(x,a,d) = \frac{1}{\varphi(d)} \int_2^x \frac{1}{\ln t}\,dt + O(x^{1/2+\epsilon})\quad\mbox{ as } \ x\to\infty <math>
where φ(d) denotes Euler's phi function and O is the Landau symbol. This is a considerable strengthening of the prime number theorem.
If GRH is true, then for every prime p there exists a primitive root modulo p (a generator of the multiplicative group of integers modulo p) which is less than 70 (ln(p))2; this is often used in proofs.
Goldbach's weak conjecture also follows from the generalized Riemann hypothesis.
If GRH is true, then the Miller-Rabin primality test is guaranteed to run in polynomial time. (A polynomial-time primality test which doesn't require GRH has recently been published; see prime number.)
Extended Riemann hypothesis (ERH)
Suppose K is a number field (a finite-dimensional field extension of the rationals Q) with ring of integers OK (this ring is the integral closure of the integers Z in K). If a is an ideal of OK, other than the zero ideal we denote its norm by Na. The Dedekind zeta function of K is then defined by
- <math>
\zeta_K(s) = \sum_a \frac{1}{(Na)^s} <math> for every complex number s with real part > 1. The sum extends over all non-zero ideals a of OK.
The Dedekind zeta function satisfies a functional equation and can be extended by analytic continuation to the whole complex plane. The resulting function encodes important information about the number field K. The extended Riemann hypothesis asserts that for every number field K and every complex number s with ζK(s) = 0: if the real part of s is between 0 and 1, then it is in fact 1/2.
The ordinary Riemann hypothesis follows from the extended one if one takes the number field to be Q, with ring of integers Z.