## Department of Mathematics

Indian Institute Of Technology Madras , Chennai

### The Prime Number Theorem

#### Abstract :

The Prime Number Theorem is arguably the most celebrated theorem in Number Theorem. It was proved in 1896 by the French mathematician J Hadamard and independently by the Belgian C de la Vallee Poussin in 1896. Both of them showed that the Riemann Zeta function has no zeros on the line ${z | Re z =1 }$ and deduced from this the assymptotic behaviour of $pi (x)$ (the number of primes less than or equal to $x$) viz. that $pi (x)/(x/log x)$ tends to 1 as $x$ tends to infinity. This last deduction was rather complicated. A much pleasanter proof was devised by D J Newman in 1980. In this talk we will present a proof of the non-vanishing of the Zeta function on $Re z =1$ followed by Newman's simple deduction of the Prime number Theorem from it.

Key Speaker Prof. M S Raghunathan