Editing Riemann zeta function (section)

===Mellin-type integrals===
The [[Mellin transform]] of a function {{math|''f''(''x'')}} is defined as<ref>{{cite journal |last=Riemann| first=Bernhard |title=[[On the number of primes less than a given magnitude]]|year=1859|journal=Monatsberichte der Königlich Preußischen Akademie der Wissenschaften zu Berlin}} translated and reprinted in {{cite book|last=Edwards|first=H. M. |authorlink=Harold Edwards (mathematician) |year=1974 |title=Riemann's Zeta Function |publisher=Academic Press |location=New York |isbn=0-12-232750-0 |zbl=0315.10035}}</ref>

:<math> \int_0^\infty f(x)x^s\, \frac{\mathrm{d}x}{x} </math>

in the region where the integral is defined. There are various expressions for the zeta function as Mellin transform-like integrals. If the real part of {{mvar|s}} is greater than one, we have

:<math>\Gamma(s)\zeta(s) =\int_0^\infty\frac{x^{s-1}}{e^x-1} \,\mathrm{d}x \quad</math> and <math>\quad\Gamma(s)\zeta(s) =\frac1{2s}\int_0^\infty\frac{x^{s}}{\cosh(x)-1} \,\mathrm{d}x</math>,

where {{math|Γ}} denotes the [[gamma function]]. By modifying the [[Contour integration|contour]], Riemann showed that

:<math>2\sin(\pi s)\Gamma(s)\zeta(s) =i\oint_H \frac{(-x)^{s-1}}{e^x-1}\,\mathrm{d}x </math>

for all {{mvar|s}}<ref>Trivial exceptions of values of {{mvar|s}} that cause removable singularities are not taken into account throughout this article.</ref> (where {{mvar|H}} denotes the [[Hankel contour]]).

We can also find expressions which relate to prime numbers and the [[prime number theorem]]. If {{math|''π''(''x'')}} is the [[prime-counting function]], then

:<math>\ln \zeta(s) = s \int_0^\infty \frac{\pi(x)}{x(x^s-1)}\,\mathrm{d}x,</math>

for values with {{math|Re(''s'') > 1}}.

A similar Mellin transform involves the Riemann function {{math|''J''(''x'')}}, which counts prime powers {{math|''p''<sup>''n''</sup>}} with a weight of {{math|{{sfrac|1|''n''}}}}, so that

: <math>J(x) = \sum \frac{\pi\left(x^\frac{1}{n}\right)}{n}.</math>

Now

:<math>\ln \zeta(s) = s\int_0^\infty J(x)x^{-s-1}\,\mathrm{d}x. </math>

These expressions can be used to prove the prime number theorem by means of the inverse Mellin transform. Riemann's [[prime-counting function]] is easier to work with, and {{math|''π''(''x'')}} can be recovered from it by [[Möbius inversion formula|Möbius inversion]].