Title: Large sums of Hecke eigenvalues of holomorphic cusp forms,
Abstract: Let $f$ be a Hecke cusp form of weight $k$ for the full modular group, and
let $\{\lambda_f(n)\}_{n\geq 1}$ be the sequence of its normalized Fourier
coefficients. Motivated by the problem of the first sign change of
$\lambda_f(n)$, we investigate the range of $x$ (in terms of $k$) for which
there are cancellations in the sum $S_f(x)=\sum_{n\leq x} \lambda_f(n)$. We
first show that $S_f(x)=o(x\log x)$ implies that $\lambda_f(n)<0$ for some
$n\leq x$. We also prove that $S_f(x)=o(x\log x)$ in the range $\log x/\log\log
k\to \infty$ assuming the Riemann hypothesis for $L(s, f)$, and furthermore
that this range is best possible unconditionally. More precisely, we establish
the existence of many Hecke cusp forms $f$ of large weight $k$, for which
$S_f(x)\gg_A x\log x$, when $x=(\log k)^A.$ Our results are $GL_2$ analogues of
work of Granville and Soundararajan for character sums, and could also be
generalized to other families of automorphic forms.