Euler’s Product Form of Riemann Zeta Function

Riemann zeta function is a rather simple-looking function. For any number s, the zeta function \zeta(s) is the sum of the reciprocals of all natural numbers raised to the s^\mathrm{th} power.

\begin{array}{ccl}\zeta(s)&=&\displaystyle\sum_{n=1}^{\infty}{\frac{1}{n^s}}\\&=&1+\frac{1}{2^s}+\frac{1}{3^s}+\frac{1}{4^s}+\frac{1}{5^s}+\frac{1}{6^s}+\cdots\end{array}

Although the variable s is a complex number, for the sake of simplicity, we will treat s as real.  (Real numbers are a subset of complex numbers.)

Continue reading “Euler’s Product Form of Riemann Zeta Function”

Advertisements