On the asymptotics of the shifted sums of Hecke eigenvalue squares

DOI10.1515/FORUM-2020-0359arXiv2011.06142MaRDI QIDQ6353566

Jiseong Kim

Publication date: 11 November 2020

Abstract: The purpose of this paper is to obtain asymptotics of shifted sums of Hecke eigenvalue squares on average. We show that for $X^{frac23+epsilon}<H<X^{1-epsilon},$ there are constants $B_h$ such that ${\displaystyle }$ sum_{Xleq n leq 2X} lambda_{f}(n)^{2}lambda_{f}(n+h)^{2}-B_{h}X=O_{f,A,epsilon}�ig(X (log X)^{-A}�ig)${\displaystyle }$ for all but $$ integers $hin[1,H]$ where ${lambd{a}_f(n)}_{ngeq1}$ are normalized Hecke eigenvalues of a fixed holomorphic cusp form $f.$ Our method is based on the Hardy-Littlewood circle method. We divide the minor arcs into two parts $m_1$ and $m_2.$ In order to treat $m_2,$ we use the Hecke relations, a bound of Miller to apply some arguments from a paper of Matom"{a}ki, Radziwill and Tao. We apply Parseval's identity and Gallagher's lemma so as to treat $m_1.$

Mathematics Subject Classification ID

Asymptotic results on arithmetic functions (11N37) Applications of the Hardy-Littlewood method (11P55) Fourier coefficients of automorphic forms (11F30)

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