Speaker: Pieter Moree (Max Planck Institute for Mathematics, Bonn, Germany)
Title: Euler-Kronecker constants and cusp forms
Date: 25/06/2021 at 10:30

Abstract: 
Ramanujan, in a manuscript not published during his life time, made a very precise conjecture 
for how many integers $n\le x$ the Ramanujan tau function $\tau(n)$ is not divisible by 691 
(and likewise for some other primes). The $\tau(n)$ are the Fourier coefficients of the Delta
function, which is a cusp form for the full modular group. Rankin proved that Ramanujan's claim 
is asymptotically correct. However, the speaker showed in 2004 that the second order behavior 
predicted by Ramanujan does not match reality. The proof makes use of high precision computation 
of constants akin to the Euler-Mascheroni constant called Euler-Kronecker constants.
   Recently the author, joint with Ciolan and Languasco, studied the analogue of Ramanujan's 
conjecture for the exceptional primes, as classified by Serre and Swinnerton-Dyer, for the 5 
cups forms akin to the Delta function for which the space of cusp forms is 1-dimensional. The 
tool for this is a high-precision evaluation of the number of integers  $n\le x$ for which a 
precribed integer $q$ does not divide  the $k$th sum of divisors function, sharpening earlier 
work of Rankin and Scourfield. In my talk I will report on generalities on Euler-Kronecker 
constants and the above work, with ample historical material.