# An explicit estimate on the mean value of the error in the prime number theorem in intervals

Date: Tue, Oct 3, 2023

Location: PIMS, University of Lethbridge, Online

Conference: Analytic Aspects of L-functions and Applications to Number Theory

Subject: Mathematics

Class: Scientific

CRG: L-Functions in Analytic Number Theory

### Abstract:

The prime number theorem (PNT) gives us the density of primes amongst the natural numbers. We can extend this idea to consider whether we have the asymptotic number of primes predicted by the PNT in a given interval. Currently, this has only been proven for sufficiently large intervals. We can also consider whether the PNT holds for sufficiently large intervals ‘on average’. This requires estimating the mean-value of the error in the PNT in intervals. A new explicit estimate for this will be given based on the work of Selberg in 1943, along with two applications: one for primes in intervals, and one for Goldbach numbers in intervals.