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<div style="font-size:30px">'''RIEMANNZETA(s)'''</div><br/>

<div style="font-size:30px">'''RIEMANNZETA(s)'''</div><br/>

*<math>s</math> is the value from 0 10 infinity.

*<math>s</math> is the value from 0 to infinity.

==Description==

==Description==

*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)=<math> \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)=\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:

*For  special values: