Manuals/calci/RIEMANNZETA

RIEMANNZETA(s)

• $s$ is the value from 0 to 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 $\zeta (s)$.
• 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.

• (ii)For negative integers:

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

• 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