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 $\scriptstyle s\,$ by the following infinite series, which converges for $\scriptstyle s\,>\,1\,$

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

This series converges over the complex plane for $\scriptstyle {\mathfrak {R}}(s)\,>\,1\,$. The function $\scriptstyle \zeta (s)\,$ can be analytically continued (holomorphically extended) to the whole $\scriptstyle s\,$-plane (except at $\scriptstyle s\,=\,1\,$,) 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

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

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

Closed form formula for even positive integers

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

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

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

No closed form formula for odd $\scriptstyle s\,$ has yet been found! $\scriptstyle \zeta (3)\,$ 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 $\scriptstyle \zeta (s)\,$ for integer $\scriptstyle s\,$ 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

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

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

Euler's alternating zeta function

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

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

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

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

$\phi (s)=(1-2^{1-s})\zeta (s),\quad s\neq 1.\,$

Some values of $\scriptstyle \phi (s)\,$

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

Riemann zeta function

Main article page: Riemann zeta function

Dirichlet L-functions

Main article page: Dirichlet L-functions

See also

• Prime Zeta Function
• Hurwitz Zeta Function
• Multivariate Zeta Function

• Euler products

External links

• Friedrich Hirzebruch, Eulerian polynomials, Münster J. of Math. 1 (2008), 9–14.