Manuals/calci/RIEMANNZETA

RIEMANNZETA(s)

$s$ is the value from 0 10 infinity.

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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \zeta(s)} .
This function is defined as the infinite series Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \zeta(s)} is a function of a complex variable Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s = \sigma + it} ,where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 :Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \zeta(s)=\sum_{n=1}^\infty n^{-s}= \frac{1}{1^s}+\frac{1}{2^s}+\frac{1}{3^s}+.....} where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma} =Real part of s>1.
We can define this by integral also:Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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.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