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

TL;DR

This paper analyzes the asymptotic behavior of divisor sums related to cusp forms, compares Landau and Ramanujan approximations, and applies findings to non-divisibility of Fourier coefficients, involving extensive computational number theory.

Contribution

It introduces a method to determine when Ramanujan's approximation outperforms Landau's for divisor sums, based on Euler-Kronecker constants and computational analysis.

Findings

Identifies pairs (k,q) where Ramanujan's approximation is superior.

Computes Euler-Kronecker constants for specific subfields.

Extends non-divisibility results of Fourier 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 $\sigma_k(n),$ the $k$-th divisor sum function. By computing the associated Euler-Kronecker constant $\gamma_{k,q},$ which depends on the arithmetic of certain subfields of $\mathbb Q(\zeta_q)$, 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 $\gamma_{k,q}<1/2$ 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.