On pairwise maxima of powers of Hecke eigenvalues
Volume 192 / 2020
Acta Arithmetica 192 (2020), 249-257
MSC: Primary 11N37; Secondary 11F30.
DOI: 10.4064/aa180721-7-2
Published online: 21 November 2019
Abstract
Let $f$ be a normalized Hecke eigenform of weight $k$ for the full modular group $\varGamma =\text {SL}(2,\mathbb {Z})$. Denote by $\lambda _f(n)$ the $n$th normalized Fourier coefficient of $f$. Let $\beta $ be a given positive real number. We establish \[ \sum _{n \leq x}\max \{|\lambda _f(n)|^{2\beta }, |\lambda _f(n+h)|^{2\beta }\}= (2a_{2\beta }+o(1))x (\log x)^{A_{2\beta }-1}, \] where $a_{2\beta }$ are suitable constants and $A_{2\beta }= \frac {4^{\beta }\varGamma (\beta +1/2)}{\sqrt {\pi }\,\varGamma (\beta +2)}.$ When $\beta \in \mathbb {Z}^+$, a stronger result is given.
