Lethbridge Number Theory and Combinatorics Seminar: Jean-Marc Deshouillers

  • Date: 04/09/2018
  • Time: 12:00
Lecturer(s):
Jean-Marc Deshouillers, University of Bordeaux
Location: 

University of Lethbridge

Topic: 

Values of arithmetic functions at consecutive arguments

Description: 

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.

Organizers:

 

Other Information: 

Time: 12:00-12:50pm

Location: B543 University Hall

Web page: http://www.cs.uleth.ca/~nathanng/ntcoseminar/