Large sums of symmetric power coefficients of holomorphic cusp forms

Abstract

Given a non-CM primitive cusp form f of even weight k and level N, we let $sy{m}^{m}f$ denote the m-th symmetric power lift of f. We denote by ${\left\{{\lambda }_{sy{m}^{m}f}\right(n\left)\right\}}_{n\in N}$ the sequence of normalized coefficients of the Dirichlet series associated to the L-function of $sy{m}^{m}f$. In this paper, we investigate the range of x (in terms of k and N) for which there are cancellations in the sum $S(x,sy{m}^{m}f)={\sum }_{n\le x}{\lambda }_{sy{m}^{m}f}\left(n\right)$. We first prove that $S(x,sy{m}^{2}f)=o(x{\log }^{2}x)$ implies that ${\lambda }_{sy{m}^{2}f}\left(n\right)<0$ for some $n\le x$. Assuming the Generalized Riemann Hypothesis (GRH) for $L(s,sy{m}^{m}f)$, we also show that $S(x,sy{m}^{m}f)=o_m(x{\log }^{m}x)$ in the range $\log x/\log \log kN\to \infty$ and $S(x,sy{m}^{m}f)=o_{m,ϵ}\left(x\right)$ in the range $x>{\left(kN\right)}^{ϵ}$.