# Erratic behavior of the coefficients of modular forms

Speaker:

Yuri Bilu, Université de Bordeaux

Date and Time:

Thursday, April 27, 2017 - 3:30pm to 4:30pm

Location:

Fields Institute, Room 210

Abstract:

q\prod_{n>0} (1-q^n)^{24} = \sum_{n>0} \tau(n) q^n . The classical work of Rankin implies that both inequalities |\tau(n)|<|\tau(n+1)| and |\tau(n)|>|\tau(n+1)| hold for infinitely many n. We generalize this for longer segments of consecutive values of \tau. Let k be a positive integer such that \tau(n) is not 0 for n\le k/2. (This is known to be true for all k < 10^{23}, and, conjecturally, holds for all k.) Let s be a permutation of the set {1,...,k}. Then there exist infinitely many positive integers n such that |\tau(n+s(1))|<\tau(n+s(2))|<...<|\tau(n+s(k))|.