Manuals/calci/RIEMANNZETA

RIEMANNZETA(s)

• $s$ is the value from $0$ to 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 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.

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(2n)=\frac{(-1)^{n+1}B_{2n}(2\pi)^{2n}}{2(2n)!}} 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 B_{2n}} is a Bernoulli number.

• (ii)For negative integers:

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(-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.

Examples

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

Related Videos

Zeta Function

See Also

• LEVENESTEST
• MOODSMEDIANTEST
• FRIEDMANTEST
• KSTESTNORMAL

References

• Riemann Zeta Function