Random ordering in modulus of consecutive Hecke eigenvalues of primitive forms

Abstract

Let τ(⋅) be the classical Ramanujan τ -function and let k be a positive integer such that τ(n)≠0 for 1≤n≤k/2 . (This is known to be true for k<10²³ , and, conjecturally, for all k .) Further, let σ be a permutation of the set {1,…,k} . We show that there exist infinitely many positive integers m such that |τ(m+σ(1))| <|τ(m+σ(2))| <⋯ <|τ(m+σ(k))| . We also obtain a similar result for Hecke eigenvalues of primitive forms of square-free level.