Editing Pi (section)

=== Number theory and Riemann zeta function ===
[[File:Prüfer.png|thumb|right|Each prime has an associated [[Prüfer group]], which are arithmetic localizations of the circle.  The [[L-function]]s of analytic number theory are also localized in each prime ''p''.]]
[[File:ModularGroup-FundamentalDomain.svg|thumb|right|Solution of the Basel problem using the [[Weil conjecture on Tamagawa numbers|Weil conjecture]]: the value of {{math|''ζ''(2)}} is the [[Poincaré half-plane model|hyperbolic]] area of a fundamental domain of the [[modular group]], times {{math|{{pi}}/2}}.]]
The [[Riemann zeta function]] {{math|''ζ''(''s'')}} is used in many areas of mathematics. When evaluated at {{math|1=''s'' = 2}} it can be written as

<math display=block>\zeta(2) = \frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} + \cdots</math>

Finding a [[closed-form expression|simple solution]] for this infinite series was a famous problem in mathematics called the [[Basel problem]]. [[Leonhard Euler]] solved it in 1735 when he showed it was equal to {{math|π<sup>2</sup>/6}}.{{sfn|Posamentier|Lehmann|2004|p=284}} Euler's result leads to the [[number theory]] result that the probability of two random numbers being [[relatively prime]] (that is, having no shared factors) is equal to {{math|6/π<sup>2</sup>}}.<ref>{{harvnb|Arndt|Haenel|2006|pp=41–43}}. {{pb}} This theorem was proved by [[Ernesto Cesàro]] in 1881. For a more rigorous proof than the intuitive and informal one given here, see {{cite book |last=Hardy |first=G. H. |author-link=G. H. Hardy |title=An Introduction to the Theory of Numbers |publisher=Oxford University Press |year=2008 |isbn=978-0-19-921986-5 |at=Theorem 332}}</ref> This probability is based on the observation that the probability that any number is [[divisible]] by a prime {{math|''p''}} is {{math|1/''p''}} (for example, every 7th integer is divisible by 7.) Hence the probability that two numbers are both divisible by this prime is {{math|1/''p''<sup>2</sup>}}, and the probability that at least one of them is not is {{math|1&nbsp;−&nbsp;1/''p''<sup>2</sup>}}. For distinct primes, these divisibility events are mutually independent; so the probability that two numbers are relatively prime is given by a product over all primes:<ref>{{cite book |author1-link=C. Stanley Ogilvy |last1=Ogilvy |first1=C. S. |last2=Anderson |first2=J. T. |title=Excursions in Number Theory |publisher=Dover |year=1988 |pages=29–35 |isbn=0-486-25778-9}}</ref>

<math display=block>\begin{align}
\prod_p^\infty \left(1-\frac{1}{p^2}\right) &= \left( \prod_p^\infty \frac{1}{1-p^{-2}} \right)^{-1}\\[4pt]
&= \frac{1}{1 + \frac{1}{2^2} + \frac{1}{3^2} + \cdots }\\[4pt]
&= \frac{1}{\zeta(2)} = \frac{6}{\pi^2} \approx 61\%.
\end{align}</math>

This probability can be used in conjunction with a [[random number generator]] to approximate {{pi}} using a Monte Carlo approach.<ref>{{harvnb|Arndt|Haenel|2006|p=43}}.</ref>

The solution to the Basel problem implies that the geometrically derived quantity {{pi}} is connected in a deep way to the distribution of prime numbers.  This is a special case of [[Weil's conjecture on Tamagawa numbers]], which asserts the equality of similar such infinite products of ''arithmetic'' quantities, localized at each prime ''p'', and a ''geometrical'' quantity: the reciprocal of the volume of a certain [[locally symmetric space]].  In the case of the Basel problem, it is the [[hyperbolic 3-manifold]] {{math|[[SL2(R)|SL<sub>2</sub>('''R''')]]/[[modular group|SL<sub>2</sub>('''Z''')]]}}.<ref>{{cite book |title=Algebraic Groups and Number Theory |first1=Vladimir |last1=Platonov |author-link1=Vladimir Platonov |first2=Andrei |last2=Rapinchuk |publisher=Academic Press |year=1994 |pages=262–265}}</ref>

The zeta function also satisfies Riemann's functional equation, which involves {{pi}} as well as the gamma function:

<math display=block>\zeta(s) = 2^s\pi^{s-1}\ \sin\left(\frac{\pi s}{2}\right)\ \Gamma(1-s)\ \zeta(1-s).</math>

Furthermore, the derivative of the zeta function satisfies

<math display=block>\exp(-\zeta'(0)) = \sqrt{2\pi}.</math>

A consequence is that {{pi}} can be obtained from the [[functional determinant]] of the [[harmonic oscillator]].  This functional determinant can be computed via a product expansion, and is equivalent to the Wallis product formula.<ref>{{cite journal |last=Sondow |first=J. |title=Analytic continuation of Riemann's zeta function and values at negative integers via Euler's transformation of series |journal=[[Proceedings of the American Mathematical Society]] |volume=120 |issue=2 |pages=421–424 |year=1994 |doi=10.1090/s0002-9939-1994-1172954-7 |citeseerx=10.1.1.352.5774 |s2cid=122276856}}</ref>  The calculation can be recast in [[quantum mechanics]], specifically the [[Calculus of variations|variational approach]] to the [[Bohr model|spectrum of the hydrogen atom]].<ref>{{cite journal |doi=10.1063/1.4930800 |first=T. |last=Friedmann |first2=C. R. |last2=Hagen |title=Quantum mechanical derivation of the Wallis formula for pi |journal=Journal of Mathematical Physics |volume=56 |issue=11 |pages=112101 |year=2015 |arxiv=1510.07813 |bibcode=2015JMP....56k2101F |s2cid=119315853}}</ref>