# Manuals/calci/RIEMANNZETA

RIEMANNZETA(s)

• is the value from to .

## 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 .
• This function is defined as the infinite series .
• 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 is a function of a complex variable ,where 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 : where =Real part of s>1.
• We can define this by integral also:
• 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.
where is a Bernoulli number.
• (ii)For negative integers:
• 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.

## Examples

1. =RIEMANNZETA(0)= -1/2
2. =RIEMANNZETA(4) = π^4/90
3. =RIEMANNZETA(-25)= -657931/12.

Zeta Function