On the coefficients of cusp forms
Fix a cuspidal representation \(\pi\) of \(GL(2)\) over (the adele ring of) a number field \(F\), and write \(a_v\) for the sum of the Hecke eigenvalues of \(\pi\) at a place \(v\) of \(F\) where \(\pi\) is unramified. Ramanujan's conjecture for \(\pi\) is equivalent to \(|a_v|\leq 2\) for all \(v\). It is shown that the lower Dirichlet density of the set of \(v\) with \(|a_v|\leq 2\) is at least \(0.9\) (one needs \(1\) for the conjecture), and that for \(\pi\) on \(GL(n)\) the set of \(v\) with \(|a_v|\leq n\) has such density \(\geq 1-1/n^2\) (but this is not directly related to Ramanujan's). Also, for \(n=2\), given \(\pi\), \(\epsilon>0\), \(m>1\), there exists a set \(S\) of primes of density \(1\) such that the partial \(L\)-function \(L_S(s,\pi,\text{sym}^m)\) converges on \(\text{Re} (s)> 1+\epsilon\).
scientific article; zbMATH DE number 1053510