Editing Prime number (section)

=== Zeta function and the Riemann hypothesis ===
{{Main|Riemann hypothesis}}
[[File:Riemann zeta function absolute value.png|thumb|upright=1.5|Plot of the absolute values of the zeta function, showing some of its features|alt=Plot of the absolute values of the zeta function]]
One of the most famous unsolved questions in mathematics, dating from 1859, and one of the [[Millennium Prize Problems]], is the [[Riemann hypothesis]], which asks where the [[zero of a function|zeros]] of the [[Riemann zeta function]] <math>\zeta(s)</math> are located.
This function is an [[analytic function]] on the [[complex number]]s. For complex numbers {{tmath|s}} with real part greater than one it equals both an [[series (mathematics)|infinite sum]] over all integers, and an [[infinite product]] over the prime numbers,
: <math>\zeta(s)=\sum_{n=1}^\infty \frac{1}{n^s}=\prod_{p \text{ prime}} \frac 1 {1-p^{-s}}.</math>
This equality between a sum and a product, discovered by Euler, is called an [[Euler product]].<ref>{{cite book |last=Patterson |first=S. J. |url=https://books.google.com/books?id=IdHLCgAAQBAJ&pg=PA1 |title=An introduction to the theory of the Riemann zeta-function |publisher=Cambridge University Press, Cambridge |year=1988 |isbn=978-0-521-33535-5 |series=Cambridge Studies in Advanced Mathematics |volume=14 |page=1 |doi=10.1017/CBO9780511623707 |mr=933558}}</ref> The Euler product can be derived from the fundamental theorem of arithmetic, and shows the close connection between the zeta function and the prime numbers.<ref>{{cite book | last1 = Borwein | first1 = Peter | author1-link = Peter Borwein | last2 = Choi | first2 = Stephen | last3 = Rooney | first3 = Brendan | last4 = Weirathmueller | first4 = Andrea | doi = 10.1007/978-0-387-72126-2 | isbn = 978-0-387-72125-5 | location = New York | mr = 2463715 | pages = 10–11 | publisher = Springer | series = CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC | title = The Riemann hypothesis: A resource for the afficionado and virtuoso alike | url = https://books.google.com/books?id=Qm1aZA-UwX4C&pg=PA10 | year = 2008 }}</ref>
It leads to another proof that there are infinitely many primes: if there were only finitely many,
then the sum-product equality would also be valid at {{tmath|1= s=1 }}, but the sum would diverge (it is the [[Harmonic series (mathematics)|harmonic series]] {{tmath|1+\tfrac{1}{2}+\tfrac{1}{3}+\dots}}) while the product would be finite, a contradiction.<ref>{{harvnb|Sandifer|2007}}, [https://books.google.com/books?id=sohHs7ExOsYC&pg=PA191 pp. 191–193].</ref>

The Riemann hypothesis states that the [[zero of a function|zeros]] of the zeta-function are all either negative even numbers, or complex numbers with [[real part]] equal to 1/2.<ref>{{harvnb|Borwein|Choi|Rooney|Weirathmueller|2008}}, [https://books.google.com/books?id=Qm1aZA-UwX4C&pg=PA15 Conjecture 2.7 (the Riemann hypothesis), p. 15].</ref> The original proof of the [[prime number theorem]] was based on a weak form of this hypothesis, that there are no zeros with real part equal to 1,<ref>{{harvnb|Patterson|1988}}, p. 7.</ref><ref name="bcrw18">{{harvnb|Borwein|Choi|Rooney|Weirathmueller|2008}}, [https://books.google.com/books?id=Qm1aZA-UwX4C&pg=PA18 p. 18.]</ref> although other more elementary proofs have been found.<ref>{{harvnb|Nathanson|2000}}, [https://books.google.com/books?id=sE7lBwAAQBAJ&pg=PA289 Chapter 9, The prime number theorem, pp. 289–324].</ref> The prime-counting function can be expressed by [[Riemann's explicit formula]] as a sum in which each term comes from one of the zeros of the zeta function; the main term of this sum is the logarithmic integral, and the remaining terms cause the sum to fluctuate above and below the main term.<ref>{{cite journal | last = Zagier | first = Don | author-link = Don Zagier | doi = 10.1007/bf03351556 | issue = S2 | journal = [[The Mathematical Intelligencer]] | pages = 7–19 | title = The first 50 million prime numbers | volume = 1 | year = 1977| s2cid = 37866599 }} See especially pp. 14–16.</ref> In this sense, the zeros control how regularly the prime numbers are distributed. If the Riemann hypothesis is true, these fluctuations will be small, and the
[[asymptotic distribution]] of primes given by the prime number theorem will also hold over much shorter intervals (of length about the square root of {{tmath|x}} for intervals near a number {{tmath|x}}).<ref name="bcrw18"/>