Generalized Riemann hypothesis

The Riemann hypothesisone ofmost important conjecturesmathematics. It isstatement aboutzeros ofRiemann zeta function. Various geometricalarithmetical objects can be described by so-called global L-functions, whichformally similar toRiemann zeta function. One can then asksame question aboutzerosthese L-functions, yielding various generalizations ofRiemann hypothesis. Nonethese conjectures have been proven or disproven, but many mathematicians believe thembe true.

Global L-functions can be associatedelliptic curves, number fields (in which case theycalled Dedekind zeta functions), Maass waveforms,Dirichlet characters (in which case theycalled Dirichlet L-functions). WhenRiemann hypothesisformulatedDedekind zeta functions, itknown asextended Riemann hypothesiswhen itformulatedDirichlet L-functions, itknown asgeneralized Riemann hypothesis. These two statements will be discussedmore detail below.

Tablecontents

1 Generalized Riemann Hypothesis (GRH) 1.1 ConsequencesGRH 2 Extended Riemann Hypothesis (ERH)

Generalized Riemann Hypothesis (GRH)

The generalized Riemann hypothesis was probably formulated forfirst time by Piltz1884. Likeoriginal Riemann hypothesis,has far reaching consequences aboutdistributionprime numbers.

The formal statement ofhypothesis follows. A Dirichlet character iscompletely multiplicative arithmetic function χ such that there existspositive integer kχ(n + k) = χ(n)all nχ(n) = 0 whenever gcd(n, k) > 1. If suchcharactergiven, we definecorresponding Dirichlet L-function by

for every complex number sreal part > 1. By analytic continuation, this function can be extended tomeromorphic function defined onwhole complex plane. The generalized Riemann hypothesis asserts thatevery Dirichlet character χevery complex number sL(χ,s) = 0: ifreal partsbetween 01, then itactually 1/2.

The case χ(n) = 1all n yieldsordinary Riemann hypothesis.

ConsequencesGRH

An arithmetic progression innatural numbers issetnumbers ofform a, a+d, a+2d, a+3d, ... where adnatural numbersdnon-zero. Dirichlet's theorem states that if adcoprime, then such an arithmetic progression contains infinitely many prime numbers. Let π(x,a,d) denotenumberprime numbersthis progression whichless than or equalx. Ifgeneralized Riemann hypothesistrue, thenevery adfor every ε > 0

where φ(d) denotes Euler's phi functionO isLandau symbol. This isconsiderable strengthening ofprime number theorem.

If GRHtrue, thenevery prime p there existsprimitive root modulo p (a generator ofmultiplicative groupintegers modulo p) whichless than 70 (ln(p))²; thisoften usedproofs.

Goldbach's weak conjecture also follows fromgeneralized Riemann hypothesis.

If GRHtrue, thenMiller-Rabin primality testguaranteedrunpolynomial time. (A polynomial-time primality test which doesn't require GRH has recently been published; see prime number.)

Extended Riemann Hypothesis (ERH)

Suppose K isnumber field (a finite-dimensional field extension ofrationals Q)ringintegers O_K (this ring isintegral closure ofintegers ZK). If aan idealO_K, other thanzero ideal we denote its norm by Na. The Dedekind zeta functionKthen defined by

for every complex number sreal part > 1. The sum extends over all non-zero ideals aO_K.

The Dedekind zeta function satisfiesfunctional equationcan be extended by analytic continuation towhole complex plane. The resulting function encodes important information aboutnumber field K. The extended Riemann hypothesis asserts thatevery number field Kevery complex number sζ_K(s) = 0: ifreal partsbetween 01, then itin fact 1/2.

The ordinary Riemann hypothesis follows fromextended one if one takesnumber fieldbe Q,ringintegers Z.