Random ordering in modulus of consecutive Hecke eigenvalues of primitive forms. (18th November 2018)

• Record Type: Journal Article
• Authors: Bilu, Yuri F.; Deshouillers, Jean-Marc; Gun, Sanoli; Luca, Florian
• Is Part Of: Compositio mathematica. Volume 154:Number 11(2018)
• Page Start: 2441
• Page End: 2461
• Publication Date: 2018-11-18
• Subjects: 11F30 (primary), -- 11F11, -- 11N36 (secondary); Fourier coefficients of modular forms, -- sieve, -- Sato–Tate; Mathematics -- Periodicals; 510
• DOI: 10.1112/S0010437X18007455
• Languages: English
• ISSNs: 0010-437X

Abstract

Abstract : Let $\unicode[STIX]{x1D70F}(\cdot )$ be the classical Ramanujan $\unicode[STIX]{x1D70F}$ -function and let $k$ be a positive integer such that $\unicode[STIX]{x1D70F}(n)\neq 0$ for $1\leqslant n\leqslant k/2$ . (This is known to be true for $k<10^{23}$, and, conjecturally, for all $k$ .) Further, let $\unicode[STIX]{x1D70E}$ be a permutation of the set $\{1, \ldots, k\}$ . We show that there exist infinitely many positive integers $m$ such that $|\unicode[STIX]{x1D70F}(m+\unicode[STIX]{x1D70E}(1))|<|\unicode[STIX]{x1D70F}(m+\unicode[STIX]{x1D70E}(2))|<\cdots <|\unicode[STIX]{x1D70F}(m+\unicode[STIX]{x1D70E}(k))|$ . We also obtain a similar result for Hecke eigenvalues of primitive forms of square-free level.