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. Many mathematicians believe these generalizations of the Riemann hypothesis to be true. The only cases of these conjectures which have been proven occur in the algebraic function field case (not the number field case).
Global L-functions can be associated to elliptic curves, number fields (in which case they are called Dedekind zeta-functions), Maass forms, 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 (ERH) and when it is formulated for Dirichlet L-functions, it is known as the generalized Riemann hypothesis or generalised Riemann hypothesis (GRH). These two statements will be discussed in more detail below. (Many mathematicians use the label generalized Riemann hypothesis to cover the extension of the Riemann hypothesis to all global L-functions, not just the special case of Dirichlet L-functions.)
The generalized Riemann hypothesis (for Dirichlet L-functions) was probably formulated for the first time by Adolf Piltz in 1884.[1] 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 for all n and whenever . If such a character is given, we define the corresponding Dirichlet L-function by
L(\chi,s)=
| infty | |
| \sum | |
| n=1 |
| \chi(n) | |
| ns |
for every complex number s such that . By analytic continuation, this function can be extended to a meromorphic function (only when
\chi
The case for all n yields the ordinary Riemann hypothesis.
Dirichlet's theorem states that if a and d are coprime natural numbers, then the arithmetic progression a,,,, ... contains infinitely many prime numbers. Let 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,
\pi(x,a,d)=
| 1 | |
| \varphi(d) |
| x | |
| \int | |
| 2 |
| 1 | |
| lnt |
dt+O(x1/2+\varepsilon) as x\toinfty,
\varphi
O
If GRH is true, then every proper subgroup of the multiplicative group
(Z/nZ) x
(Z/nZ) x
If GRH is true, then 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((lnp)6).
Goldbach's weak conjecture also follows from the generalized Riemann hypothesis. The yet to be verified proof of Harald Helfgott of this conjecture verifies the GRH for several thousand small characters up to a certain imaginary part to obtain sufficient bounds that prove the conjecture for all integers above 1029, integers below which have already been verified by calculation.[5]
Assuming the truth of the GRH, the estimate of the character sum in the Pólya–Vinogradov inequality can be improved to
O\left(\sqrt{q}loglogq\right)
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
\zetaK(s)=\suma
| 1 | |
| (Na)s |
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.
The ERH implies an effective version[6] 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
| |C| | |
| |G| |
l(\operatorname{li}(x)+Ol(\sqrtx(nlogx+log|\Delta|)r)r),