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# Zeta function

Zeta functions generally involve taking the reciprocals of an infinite power series. The most famous zeta function is the Riemann zeta function.

## Euler's zeta function

Leonhard Euler defined the zeta function of a real variable ${\displaystyle \scriptstyle s\,}$ by the following infinite series, which converges for ${\displaystyle \scriptstyle s\,>\,1\,}$

${\displaystyle \zeta (s)\equiv \sum _{n=1}^{\infty }{\frac {1}{n^{s}}}={\frac {1}{1^{s}}}+{\frac {1}{2^{s}}}+{\frac {1}{3^{s}}}+{\frac {1}{4^{s}}}+\cdots ,\quad s>1.\,}$

This series converges over the complex plane for ${\displaystyle \scriptstyle {\mathfrak {R}}(s)\,>\,1\,}$. The function ${\displaystyle \scriptstyle \zeta (s)\,}$ can be analytically continued (holomorphically extended) to the whole ${\displaystyle \scriptstyle s\,}$-plane (except at ${\displaystyle \scriptstyle s\,=\,1\,}$,) and this will be known as the Riemann zeta function, introduced later in 1859.

### Euler's product

He then discovered that it has a deep connection with the prime numbers and proved this famous result, known as Euler's product

${\displaystyle \zeta (s)=\prod _{i=1}^{\infty }{\frac {1}{1-{\tfrac {1}{{p_{i}}^{s}}}}}={\frac {1}{1-{\tfrac {1}{2^{s}}}}}\cdot {\frac {1}{1-{\tfrac {1}{3^{s}}}}}\cdot {\frac {1}{1-{\tfrac {1}{5^{s}}}}}\cdot {\frac {1}{1-{\tfrac {1}{7^{s}}}}}\cdot {\frac {1}{1-{\tfrac {1}{11^{s}}}}}\cdots ,\,}$

where ${\displaystyle \scriptstyle p_{i}\,}$ is the ${\displaystyle \scriptstyle i\,}$th prime.

### Closed form formula for even positive integers

Euler studied the function for integer values of ${\displaystyle \scriptstyle s\,}$ and succeeded in finding a closed form formula in terms of the Bernoulli numbers for positive even integers

${\displaystyle \zeta (2k)={\frac {2^{2k-1}~\pi ^{2k}}{(2k)!}}~|B_{2k}|={\frac {1}{2}}\,{\frac {(2\pi )^{2k}}{(2k)!}}\,|B_{2k}|,\quad k\in \mathbb {N} ^{+},\,}$

where ${\displaystyle \scriptstyle B_{n}\,}$ is the ${\displaystyle \scriptstyle n\,}$th Bernoulli number.

No closed form formula for odd ${\displaystyle \scriptstyle s\,}$ has yet been found! ${\displaystyle \scriptstyle \zeta (3)\,}$ is known as Apéry's constant. It was named for Roger Apéry (1916–1994), who in 1978 proved it to be irrational.

The values of ${\displaystyle \scriptstyle \zeta (s)\,}$ for integer ${\displaystyle \scriptstyle s\,}$ in range 2 to 12 are given in the last column of the table of related formulae and values of regular orthotopic numbers.

### Reciprocal of the zeta function

The reciprocal of the zeta function is obtained by Möbius inversion

${\displaystyle {\frac {1}{\zeta (s)}}=\sum _{n=1}^{\infty }{\frac {\mu (n)}{n^{s}}},\,}$

where ${\displaystyle \scriptstyle \mu (n)\,}$ is the Möbius function.

Equivalently, we may say that the Möbius function gives the Dirichlet generating sequence of the reciprocal of the Riemann zeta function.

Conversely, ${\displaystyle \scriptstyle {\frac {1}{\zeta (s)}}\,}$ is the Dirichlet generating function of the Möbius function ${\displaystyle \scriptstyle \mu (n)\,}$ for all positive integers.

## Euler's alternating zeta function

Euler also defined the alternating zeta function ${\displaystyle \scriptstyle \phi (s)\,}$ (also denoted ${\displaystyle \scriptstyle \zeta ^{*}(s)\,}$) of a real variable ${\displaystyle \scriptstyle s\,}$ (also known as the Dirichlet eta function ${\displaystyle \scriptstyle \eta (s)\,}$) by the following infinite series, which converges for ${\displaystyle \scriptstyle s\,>\,0\,}$

${\displaystyle \phi (s)\equiv \sum _{n=1}^{\infty }{\frac {(-1)^{n+1}}{n^{s}}}={\frac {1}{1^{s}}}-{\frac {1}{2^{s}}}+{\frac {1}{3^{s}}}-{\frac {1}{4^{s}}}+\cdots ,\quad s>0.\,}$

Euler wanted to calculate ${\displaystyle \scriptstyle \phi (s)\,}$ for ${\displaystyle \scriptstyle s\,=\,-m\,}$ with ${\displaystyle \scriptstyle m\,}$ = 0, 1, 2, 3, ... and to do so he introduced the Eulerian polynomials.

This series converges over the complex plane for ${\displaystyle \scriptstyle {\mathfrak {R}}(s)\,>\,0\,}$. The function ${\displaystyle \scriptstyle \phi (s)\,}$ can be analytically continued (holomorphically extended) to the whole ${\displaystyle \scriptstyle s\,}$-plane. It is related to the ${\displaystyle \scriptstyle \zeta \,}$-function by the relation

${\displaystyle \phi (s)=(1-2^{1-s})\zeta (s),\quad s\neq 1.\,}$
Some values of ${\displaystyle \scriptstyle \phi (s)\,}$
${\displaystyle \phi (0)=(1-2^{1-0})\zeta (0)=-(-{\tfrac {1}{2}})={\tfrac {1}{2}}\,}$
${\displaystyle \phi (1)=\sum _{n=1}^{\infty }{\frac {(-1)^{n+1}}{n}}=\log(2),{\rm {~from~}}\log(1+x)=\sum _{n=1}^{\infty }{\frac {(-1)^{n+1}}{n}}x^{n},\quad -1
${\displaystyle \phi (2)={\tfrac {1}{2}}\zeta (2)={\tfrac {1}{2}}({\tfrac {1}{6}}\pi ^{2})={\tfrac {1}{12}}\pi ^{2}\,}$
${\displaystyle \phi (3)={\tfrac {3}{4}}\zeta (3)\,}$
${\displaystyle \phi (4)={\tfrac {7}{8}}\zeta (4)={\tfrac {7}{8}}({\tfrac {1}{90}}\pi ^{4})={\tfrac {7}{720}}\pi ^{4}\,}$
${\displaystyle \phi (5)={\tfrac {15}{16}}\zeta (5)\,}$

## Riemann zeta function

Main article page: Riemann zeta function

## Dirichlet L-functions

Main article page: Dirichlet L-functions