Large Sums of Fourier Coefficients of Cusp Forms

Let $N$ be a fixed positive integer, and let $f\in S_k(N)$ be a primitive cusp form given by the Fourier expansion $f(z)=\sum_{n=1}^{\infty} \lambda_f(n)n^{\frac{k-1}{2}}e(nz)$. We consider the partial sum $S(x,f)=\sum_{n\leq x}\lambda_f(x)$. It is conjectured that $S(x,f)=o(x\log x)$ in the range $x\geq k^{\epsilon}$. 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 $\epsilon>(\log k)^{-\frac{1}{8}}$ and $1\leq T\leq (\log k)^{\frac{1}{200}}$, we have $S(x,f)\ll \frac{x\log x}{T}$ in the range $x\geq k^{\epsilon}$ provided that $L(s,f)$ has no more than $\epsilon^2\log k/5000$ zeros in the region $\left\{s\,:\, \Re(s)\geq \frac34, \, |\Im(s)-\phi| \leq \frac14\right\}$ for every real number $\phi$ with $|\phi|\leq T$.