Random ordering in modulus of consecutive Hecke eigenvalues of primitive forms
arXiv:1701.01915 · doi:10.1112/S0010437X18007455
Abstract
Let τ(.) be the Ramanujan τ-function, and let k be a positive integer such that τ(n) is not 0 for n=1,...,[k/2]. (This is known to be true for k < 10^{23}, and, conjecturally, for all k.) Further, let s be a permutation of the set {1,...,k}. Then there exist infinitely many positive integers m such that |τ(m+s(1))|<τ(m+s(2))|<...<|τ(m+s(k))|. We also obtain a similar result for Fourier-coefficients of general newforms.
Minor corrections. To appear in Compositio Math