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 $\unicode[STIX]{x1D70F}(· )$ be the classical Ramanujan $\unicode[STIX]{x1D70F}$ -function and let $k$ be a positive integer such that $\unicode[STIX]{x1D70F}(n)≠ 0$ 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 $\{1, …, k\}$ . We show that there exist infinitely many positive integers $m$ such that $|\unicode[STIX]{x1D70F}(m+\unicode[STIX]{x1D70E}(1))|<|\unicode[STIX]{x1D70F}(m+\unicode[STIX]{x1D70E}(2))|<… <|\unicode[STIX]{x1D70F}(m+\unicode[STIX]{x1D70E}(k))|$ . We also obtain a similar result for Hecke eigenvalues of primitive forms of square-free level.