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- W4298020194 abstract "Let $f$ be a Hecke cusp form of weight $k$ for the full modular group, and let ${lambda_f(n)}_{ngeq 1}$ 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 $S_f(x)=sum_{nleq x} lambda_f(n)$. We first show that $S_f(x)=o(xlog x)$ implies that $lambda_f(n)<0$ for some $nleq x$. We also prove that $S_f(x)=o(xlog x)$ in the range $log x/loglog kto infty$ 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 $S_f(x)gg_A xlog x$, 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." @default.
- W4298020194 created "2022-10-01" @default.
- W4298020194 creator A5086450144 @default.
- W4298020194 date "2017-03-30" @default.
- W4298020194 modified "2023-09-29" @default.
- W4298020194 title "Large sums of Hecke eigenvalues of holomorphic cusp forms" @default.
- W4298020194 doi "https://doi.org/10.48550/arxiv.1703.10582" @default.
- W4298020194 hasPublicationYear "2017" @default.
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