Editing Generalized Riemann hypothesis (section)

=== 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.