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

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