The Riemann Zeta Function and the Prime Number Theorem

Abstract

For a complex number s, we always denote its real part by $\sigma$ and imaginary part by t. Thus $s=\sigma +it.$ The Riemann Zeta function  is defined as

$$ \zeta (s)=\sum ^{\infty }_{n=1} \ \ \frac{1}{n^s} \ \text {in} \ \sigma > 1.$$