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

Abstract

In 1961, Rankin determined the asymptotic behavior of the number $S_{k,q}(x)$ of positive integers $n\le x$ for which a given prime $q$ does not divide $σ_k(n),$ the $k$-th divisor sum function. By computing the associated Euler-Kronecker constant $γ_{k,q},$ which depends on the arithmetic of certain subfields of $\mathbb Q(ζ_q)$, we obtain the second order term in the asymptotic expansion of $S_{k,q}(x).$ Using a method developed by Ford, Luca and Moree (2014), we determine the pairs $(k,q)$ with $(k, q-1)=1$ for which Ramanujan's approximation to $S_{k,q}(x)$ is better than Landau's. This entails checking whether $γ_{k,q}<1/2$ or not, and requires a substantial computational number theoretic input and extensive computer usage. We apply our results to study the non-divisibility of Fourier coefficients of six cusp forms by certain exceptional primes, extending the earlier work of Moree (2004), who disproved several claims made by Ramanujan on the non-divisibility of the Ramanujan tau function by five such exceptional primes.