View source for Generalized Riemann hypothesis

{{Short description|Mathematical conjecture about zeros of L-functions}}
The [[Riemann hypothesis]] is one of the most important [[conjecture]]s 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-function|''L''-function]]s, 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 curve]]s, [[number field]]s (in which case they are called [[Dedekind zeta-function]]s), [[Maass form]]s, and [[Dirichlet character]]s (in which case they are called [[Dirichlet L-series|Dirichlet L-function]]s). 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.)

== Generalized Riemann hypothesis (GRH) ==
The generalized Riemann hypothesis (for Dirichlet ''L''-functions) was probably formulated for the first time by [[Adolf Piltz]] in 1884.<ref>{{cite book |author-link=Harold Davenport |last=Davenport |first=Harold |title=Multiplicative Number Theory |edition=Third |others=Revised and with a preface by [[Hugh L. Montgomery]] |series=Graduate Texts in Mathematics |volume=74 |publisher=Springer-Verlag |location=New York |year=2000 |isbn=0-387-95097-4 |page=124 |url=https://books.google.com/books?id=SFztBwAAQBAJ&pg=PA124 }}</ref> Like the original Riemann hypothesis, it has far reaching consequences about the distribution of [[prime number]]s.

The formal statement of the hypothesis follows. A [[Dirichlet character]] is a [[multiplicative function|completely multiplicative]] [[arithmetic function]] ''χ'' such that there exists a positive integer ''k'' with {{nowrap|1=''χ''(''n'' + ''k'') = ''χ''(''n'')}} for all ''n'' and {{nowrap|1=''χ''(''n'') = 0}} whenever {{nowrap|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'' such that {{nowrap|Re ''s'' > 1}}. By [[analytic continuation]], this function can be extended to a [[meromorphic function]] (only when <math> \chi </math> is primitive) defined on the whole complex plane. The generalized Riemann hypothesis asserts that, for every Dirichlet character ''χ'' and every complex number ''s'' with {{nowrap|1=''L''(''χ'', ''s'') = 0}}, if ''s'' is not a negative real number, then the real part of ''s'' is 1/2.

The case {{nowrap|1=''χ''(''n'') = 1}} for all ''n'' yields the ordinary Riemann hypothesis.

=== Consequences of GRH ===
[[Dirichlet's theorem on arithmetic progressions|Dirichlet's theorem]] states that if ''a'' and ''d'' are [[coprime]] [[natural number]]s, then the [[arithmetic progression]] ''a'', {{nowrap|''a'' + ''d''}}, {{nowrap|''a'' + 2''d''}}, {{nowrap|''a'' + 3''d''}}, ... contains [[Infinite set|infinitely many]] prime numbers. Let {{nowrap|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 {{nowrap|''ε'' > 0}},
:<math>\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,</math>
where <math>\varphi</math> is [[Euler's totient function]] and <math>O</math> is the [[Big O notation]]. This is a considerable strengthening of the [[prime number theorem]].

If GRH is true, then every proper subgroup of the multiplicative group <math>(\mathbb Z/n\mathbb Z)^\times</math> omits a number less than {{nowrap|2(ln ''n'')<sup>2</sup>}}, as well as a number coprime to ''n'' less than {{nowrap|3(ln ''n'')<sup>2</sup>}}.<ref>{{Cite journal |last=Bach |first=Eric |year=1990 |title=Explicit bounds for primality testing and related problems |journal=[[Mathematics of Computation]] |volume=55 |issue=191 |pages=355–380 |doi=10.2307/2008811 |jstor=2008811 |doi-access=free }}</ref> In other words, <math>(\mathbb Z/n\mathbb Z)^\times</math> is generated by a set of numbers less than {{nowrap|2(ln ''n'')<sup>2</sup>}}. This is often used in proofs, and it has many consequences, for example (assuming GRH):
*The [[Miller–Rabin primality test]] is guaranteed to run in polynomial time. (A polynomial-time primality test which does not require GRH, the [[AKS primality test]], was published in 2002.)
*The [[Shanks–Tonelli algorithm]] is guaranteed to run in polynomial time.
*The Ivanyos–Karpinski–Saxena deterministic algorithm<ref>{{cite book|first1=Gabor|last1=Ivanyos|last2=Karpinski|first2=Marek|first3=Nitin|last3=Saxena|title=Proceedings of the 2009 international symposium on Symbolic and algebraic computation (ISAAC)|chapter=Schemes for deterministic polynomial factoring|year=2009|pages=191–198|doi=10.1145/1576702.1576730|isbn=9781605586090|arxiv=0804.1974|s2cid=15895636}}</ref> for factoring polynomials over finite fields with prime constant-smooth degrees is guaranteed to run in polynomial time.

If GRH is true, then for every prime ''p'' there exists a [[Primitive root modulo n|primitive root mod ''p'']] (a generator of the multiplicative group of integers modulo ''p'') that is less than <math>O((\ln p)^6).</math><ref>{{cite journal |first=Victor |last=Shoup |title=Searching for primitive roots in finite fields |journal=Mathematics of Computation |volume=58 |issue=197 |year=1992 |pages=369–380 |doi= 10.2307/2153041|jstor=2153041 |doi-access=free }}</ref>

[[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 10<sup>29</sup>, integers below which have already been verified by calculation.<ref>p5. {{cite arXiv|last=Helfgott|first=Harald|eprint=1305.2897|title=Major arcs for Goldbach's theorem|class=math.NT|year=2013}}</ref>

Assuming the truth of the GRH, the estimate of the character sum in the [[Character sum|Pólya–Vinogradov inequality]] can be improved to <math>O\left(\sqrt{q}\log\log q\right)</math>, ''q'' being the modulus of the character.

== Extended Riemann hypothesis (ERH) ==

Suppose ''K'' is a [[number field]] (a finite-dimensional [[field extension]] of the [[rational number|rationals]] <math>\mathbb Q</math>) with [[algebraic number|ring of integers]] O<sub>''K''</sub> (this ring is the [[integral closure]] of the [[integer]]s <math>\mathbb Z</math> in ''K''). If ''a'' is an [[ring ideal|ideal]] of O<sub>''K''</sub>, other than the zero ideal, we denote its [[ideal norm|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 >&nbsp;1. The sum extends over all non-zero ideals ''a'' of O<sub>''K''</sub>.

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 ζ<sub>''K''</sub>(''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 <math>\mathbb Q</math>, with ring of integers <math>\mathbb Z</math>.

The ERH implies an effective version<ref>{{cite journal|first1=J.C.|last1=Lagarias|first2=A.M.|last2=Odlyzko|title=Effective Versions of the Chebotarev Theorem|journal=Algebraic Number Fields|year=1977|pages=409–464}}</ref> 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 [[Ramification (mathematics)#In algebraic number theory|unramified primes]] of ''K'' of norm below ''x'' with Frobenius conjugacy class in ''C'' is
:<math>\frac{|C|}{|G|}\Bigl(\operatorname{Li}(x)+O\bigl(\sqrt x(n\log x+\log|\Delta|)\bigr)\Bigr),</math>
where the constant implied in the big-O notation is absolute, ''n'' is the degree of ''L'' over ''Q'', and Δ its discriminant.

== See also ==
* [[Artin conjecture (L-functions)|Artin's conjecture]]
* [[Dirichlet L-function]]
* [[Selberg class]]
* [[Grand Riemann hypothesis]]

==References==
{{reflist}}

==Further reading==
*{{Springer|id=R/r081940|title=Riemann hypothesis, generalized}}

{{Bernhard Riemann}}
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[[Category:Zeta and L-functions]]
[[Category:Algebraic geometry]]
[[Category:Conjectures]]
[[Category:Unsolved problems in mathematics]]
[[Category:Bernhard Riemann]]