Hall 123

Landau's Oscillation Theorem

Kamalakshya Mahatab

IMSc

The famous Prime Number Theorem says:Sum of log(p),for primes p,up to x is asymptotic to x.Denote A(x)=sum(log(p): p<= x).If one assumes Riemann Hypothesis then it can be shown that A(x)-x <= x^{1/2+c} for any small positive constant c.Say D(x)= A(x)-x. We may ask the following questions about D(x):
* Is x^{1/2+c} is a necessary upper bound for D(x)?
* Is D(x) always positive or always negative for large enough x?
Answers to the above two questions can be given using Landau's theorem,which will be the focus of this talk. This talk requires no expertise in analytic number theory.Basic knowledge of real and complex analysis will be sufficient to follow the talk.