Manuals/calci/RIEMANNZETA

RIEMANNZETA(s)

Description

This function gives the result for the function of Riemann-Zeta function.
It is also known as Euler-Riemann Zeta function.
This function is useful in number theory for the investigating properties of prime numbers.
It is denoted by $\zeta (s)$ .
This function is defined as the infinite series $\zeta (s)=1+2^{-s}+3^{-s}+.....$ .
When the value of s=1,then this series is called the harmonic series.
When it is increase without any bound or limit, then its sum is infinite.
When the value of s is larger than 1,the the series converges to a finite number as successive terms are added.
The riemann zeta function is defined for Complex numbers also.
So $\zeta (s)$ is a function of a complex variable $s=\sigma +it$ ,where $\sigma$ and t are real numbers.i is the imaginary unit.
It is also a function of a complex variable s that analytically continues the sum of the infinite series , which converges when the real part of s is greater than 1.
It is defined by : $\zeta (s)=\sum _{n=1}^{\infty }n^{-s}={\frac {1}{1^{s}}}+{\frac {1}{2^{s}}}+{\frac {1}{3^{s}}}+.....$ where $\sigma$ =Real part of s>1.
We can define this by integral also: $\zeta (s)={\frac {1}{\Gamma (s)}}\int \limits _{0}^{\infty }{\frac {x^{s-1}}{e^{x-1}}}\,dx$
Riemann zeta function is a meromorphic function on the whole complex s-plane, which is holomorphic everywhere except for a simple pole at s = 1 with residue 1.
For special values:
(i)Any positive integer 2n.

\zeta (2n)={\frac {(-1)^{n+1}B_{2n}(2\pi )^{2n}}{2(2n)!}}

where

B_{2n}

is a Bernoulli number.

\zeta (-n)={\frac {-B_{n+1}}{n+1}}

For n ≥ 1,so in particular ζ vanishes at the negative even integers because Bm = 0 for all odd m other than 1.
For odd positive integers, no such simple expression is known.
- When s=1,then ζ (1) is Harmonic series.
- when s=2, then ζ (2) derivation is Basel problem.
- when s=3, then ζ (3) derivation is Apery's constant.
- When s=4, then ζ (4) derivation is Planck's law.