Editing Gamma function (section)

=== Analytic number theory ===

An application of the gamma function is the study of the [[Riemann zeta function]]. A fundamental property of the Riemann zeta function is its [[functional equation]]:
<math display="block">\Gamma\left(\frac{s}{2}\right)\zeta(s)\pi^{-\frac{s}{2}} = \Gamma\left(\frac{1-s}{2}\right)\zeta(1-s)\pi^{-\frac{1-s}{2}}.</math>

Among other things, this provides an explicit form for the [[analytic continuation]] of the zeta function to a meromorphic function in the complex plane and leads to an immediate proof that the zeta function has infinitely many so-called "trivial" zeros on the real line. Borwein ''et al.'' call this formula "one of the most beautiful findings in mathematics".<ref>{{cite book |author = Borwein, J. |author2 = Bailey, D. H. |author3 = Girgensohn, R. |name-list-style = amp |year = 2003 |title = Experimentation in Mathematics |publisher = A. K. Peters |pages = 133 |isbn = 978-1-56881-136-9 }}</ref> Another contender for that title might be
<math display="block">\zeta(s) \; \Gamma(s) = \int_0^\infty \frac{t^s}{e^t-1} \, \frac{dt}{t}.</math>

Both formulas were derived by [[Bernhard Riemann]] in his seminal 1859 paper "''[[On the Number of Primes Less Than a Given Magnitude|Ueber die Anzahl der Primzahlen unter einer gegebenen Größe]]''" ("On the Number of Primes Less Than a Given Magnitude"), one of the milestones in the development of [[analytic number theory]]—the branch of mathematics that studies [[prime number]]s using the tools of mathematical analysis.