Lethbridge Number Theory and Combinatorics Seminar: Jean-Marc Deshouillers

We shall place in a general context the following result recently (*) obtained jointly with Yuri Bilu (Bordeaux), Sanoli Gun (Chennai) and Florian Luca (Johannesburg).

Theorem. 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))∣∣.

The proof uses sieve method, Sato-Tate conjecture, recurrence relations for the values of τ at prime power values.

(*) Hopefully to appear in 2018.