Landau and Ramanujan approximations for divisor sums and coefficients of cusp forms
Alexandru Ciolan, Alessandro Languasco, Pieter Moree
TL;DR
The paper addresses the second-order asymptotics for the counting function S_{k,q}(x) = ∑_{n≤x, q ∤ σ_k(n)} 1, tying it to Euler-Kronecker constants γ_S of cyclotomic subfields. It develops a Dirichlet-series framework T(s) for the indicator of q ∤ σ_k(n), expresses γ_{k,q} in terms of γ and Dedekind zeta-constants of K_r and K_{2r}, and gives precise S_{k,q}(x) and S'_{k,q}(x) expansions with δ_q = 1/h where h=(q-1)/r and r=(k,q-1). A Ramanujan-type refinement is shown to be valid only when γ_{k,q} < 1/2, and generically, for large primes q with (k,q-1)=1, the Landau approximation dominates due to γ_{k,q} ≈ γ > 1/2, with explicit exceptional-prime cases analyzed for cusp forms. The paper also applies these results to non-divisibility of Fourier coefficients of six cusp forms by exceptional primes, provides detailed numerical computations, and discusses broad implications for Ramanujan-type congruences and potential generalizations.
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.