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arXiv · 2110.06855

Applications of zero-free regions on averages and shifted convolution sums of Hecke eigenvalues

Jiseong Kim
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Abstract

By assuming Vinogradov-Korobov type zero-free regions and the generalized Ramanujan-Petersson conjecture, we establish nontrivial upper bounds for almost all short sums of Fourier coefficients of Hecke-Maass cusp forms for $SL(n,\mathbb{Z})$. As applications, we obtain nontrivial upper bounds for the averages of shifted sums involving coefficients of the Hecke-Maass cusp forms for $SL(n,\mathbb{Z})$. Furthermore, we present a conditional result regarding sign changes of these coefficients.

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Jiseong Kim. 2021-10-13. Applications of zero-free regions on averages and shifted convolution sums of Hecke eigenvalues. https://arxiv.org/abs/2110.06855

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