Euler-Kronecker constants and cusp forms

Abstract: Ramanujan, in a manuscript not published during his lifetime, 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 prescribed 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 of historical material.

number theory

Audience: researchers in the discipline

Dublin Algebra and Number Theory Seminar

Series comments: Passcode: The 3-digit prime numerator of Riemann zeta at -11