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*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 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 :<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.
*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:<math>zeta(s)=\frac{1}{\Gamma(s)}\int\limits_{0}^{\infty} \frac{x^{s-1}{e^{x-1}\,dx </math>
*We can define this by integral also:<math>zeta(s)=<math> \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.
:<math>\zeta(2n)=\frac{(-1)^{n+1}B_{2n}(2\pi)^{2n}}{2(2n)!}</math> where <math>B_{2n}</math> is a Bernoulli number.
*(ii)For negative integers:
:<math>\zeta(-n)=\frac{-B_{n+1}}{n+1}</math>
*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.
*For odd positive integers, no such simple expression is known.
**When s=1,then ζ (1) is Harmonic series.
**When s=1,then ζ (1) is Harmonic series.