ULethbridge Number Theory and Combinatorics Seminar: Andrew Fiori

In recent work with Mikko Jaskari we study the problem of finding tight explicit bounds on sums over zeros of L-functions near the 1-line with a variety of zero-free regions and zero-density estimates. As an application to demonstrate the techniques, we apply them to the case of the Riemann zeta function and the prime number theorem to obtain new best estimates on the error terms in the prime number theorem for large x.
We show that for log(x)≥3
we have
|ψ(x)−x|<0.239xexp(−0.1982767(1+logloglog(x)15loglog(x))log(x)3/5loglog(x)1/5)
and for log(x)>1015
we have
|ψ(x)−x|<(1.1485×10−1154556)xexp(−0.1982767(1+logloglog(x)15loglog(x))log(x)3/5loglog(x)1/5).
In this talk I will explain how we obtain this result.