Large Sums of Fourier Coefficients of Cusp Forms

Abstract

Let $N$ be a fixed positive integer, and let $f\in S_k(N)$ be a primitive cusp form given by the Fourier expansion $f(z)={\sum }_{n=1}^{\infty }\hat{I}{»}_f(n)n^{\frac{k-1}{2}}e(nz)$. We consider the partial sum $S(x,f)={\sum }_{n\le x}\hat{I}{»}_f(x)$. It is conjectured that $S(x,f)=o(x\log x)$ in the range $x\ge k^{\hat{I}}\mu$. Lamzouri proved in arXiv:1703.10582 [math.NT] that this is true under the assumption of the Generalized Riemann Hypothesis (GRH) for $L(s,f)$. In this paper, we prove that this conjecture holds under a weaker assumption than GRH. In particular, we prove that given $\hat{I}\mu >(\log k{)}^{-\frac{1}{8}}$ and $1\le T\le (\log k{)}^{\frac{1}{200}}$, we have $S(x,f)\ll \frac{x\log x}{T}$ in the range $x\ge k^{\hat{I}}\mu$ provided that $L(s,f)$ has no more than $\hat{I}{\mu }^2\log k/5000$ zeros in the region $\left\{s:\Re (s)\ge \frac{3}{4},|\Im (s)-\ddot{I}•|\le \frac{1}{4}\right\}$ for every real number $\ddot{I}•$ with $|\ddot{I}•|\le T$.

14 pages