The first negative coefficients of symmetric square \(L\)-functions (Q427280)

Denote by \(\lambda_{\text{sym}^2 f}(n)\) the \(n\)th coefficient of the Dirichlet series representation of the symmetric square \(L\)-function \(L(s, \text{sym}^2 f)\) associated to a holomorphic primitive cusp form \(f\) of level \(N\) and weight \(k\). The authors of the present paper look for a bound for the first sign change of the sequence \(\lambda_{\text{sym}^2 f}(n)\). More precisely, writing \(n_{\text{sym}^2 f}\) for the smallest positive integer \(n\) such that \((n,N) = 1\) and \(\lambda_{\text{sym}^2} f(n) < 0\), it is shown that \(n_{\text{sym}^2 f} \ll (k^2 N^2)^{40/113}\). The proof uses a set-up similar to what has been used in earlier works on sign changes of \(\text{GL}_2\) modular forms, that is, the authors give upper and lower bounds for the sum \[ \sum _{\substack{ n \leq y^u \\ (n,N)=1 }} \mu(n)^2 \lambda_{\text{sym}^2 f} (n), \] where \(u > 1\) and \(y = n_{\text{sym}^2 f}\). The needed upper bound follows from the convexity bound for the symmetric square \(L\)-function whereas the main ingredients in the proof of the lower bound are studying the size of individual coefficients \(\lambda_{\text{sym}^2 f}(p)\) in different ranges for prime \(p\) and then using a mean-value result for multiplicative functions (proved in the paper). Some new arguments compared to the \(\text{GL}_2\) case are needed in the proof of the lower bound.