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*This function is useful in number theory for the investigating properties of prime numbers.  

*This function is useful in number theory for the investigating properties of prime numbers.  

*It is denoted by <math>\zeta(s)</math>.  

*It is denoted by <math>\zeta(s)</math>.  

*This function is defined as the infinite series ζ(s)=1+2^-s+3^-s+.....  

*This function is defined as the infinite series <math>\zeta(s)=1+2^{-s}+3^{-s}+.....</math>.  

*When the value of s=1,then this series is called the harmonic 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 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.  

*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.  

*The riemann zeta function is defined for Complex numbers also.  

*So  ζ(s) is a function of a complex variable s = σ + it.  

*So  <math>\zeta(s)</math> is a function of a complex variable <math>s = \sigma + it</math>,where <math>\sigma</math> and t are real numbers.i is the imaginary unit.  

*It is 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 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 :ζ(s)=summation n= 1 to infinity n^-s= 1/1^s+1/2^s+1/3^s+..... where σ =Real part of s>1.

*It is defined by :<math>\zeta(s)=\sum_{n=1}^\infty n^{-s}= \frac{1}{1^s}+\frac{1}{2^s}+\frac{1}{3^s}+.....</math> where <math>\sigma</math> =Real part of s>1.

*We can define this by integral also:ζ(s)=1/gamma (s)integral 0 to infinity x^(s-1)/e^x-1 dx.

*We can define this by integral also:<math>zeta(s)=\frac{1}{\Gamma(s)}\int\limits_{0}^{\infty} \frac{x^{s-1}{e^{x-1}\,dx </math>

*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.

*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  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.