Random ordering in modulus of consecutive Hecke eigenvalues of primitive forms. (18th November 2018) - Bilu, Yuri F.; Deshouillers, Jean-Marc; Gun, Sanoli; Luca, Florian
Abstract : Let [STIX]x1D70F(· ) be the classical Ramanujan [STIX]x1D70F -function and let k be a positive integer such that [STIX]x1D70F(n)≠ 0 for 1nk/2 . (This is known to be true for k<10^23, and, conjecturally, for all k .) Further, let [STIX]x1D70E be a permutation of the set {1, …, k} . We show that there exist infinitely many positive integers m such that |[STIX]x1D70F(m+[STIX]x1D70E(1))|<|[STIX]x1D70F(m+[STIX]x1D70E(2))|<… <|[STIX]x1D70F(m+[STIX]x1D70E(k))| . We also obtain a similar result for Hecke eigenvalues of primitive forms of square-free level.