Landau and Ramanujan approximations for divisor sums and coefficients of cusp forms

Zoom ID: 919 6497 4060. For password please contact Pieter Moree (moree@mpim...).

In his "unpublished" manuscript, Ramanujan made a few claims about the asymptotic behavior of the number of positive integers $n\le x$ for which a prime $q\in\{2,3,5,7,23,691\}$ does not divide $\tau(n),$ claims which were disproven by Moree (2004). Due to the work of Deligne, Serre and Swinnerton-Dyer, we know that these primes are only a few out of a much larger list of exceptional primes modulo which $\tau(n)$ and the coefficients of five other cusp forms satisfy congruences involving $\sigma_k(n).$
In this talk, I will report on an ongoing work, joint with Alessandro Languasco and Pieter Moree, in which we study the number of positive integers $n\le x$ such that $q\nmid n^a\sigma_k(n),$ for a given prime $q.$  Using computational number theory and extensive numerical calculations, we determine in each case whether the Landau or the Ramanujan  approximation is a better estimate for this quantity. In addition, by computing the associated Euler-Kronecker constants, we show that the Ramanujan-type claims are false for all the six cusp forms and the exceptional primes. This considerably extends and puts into a general framework the earlier work of Moree (2004).