Number Theory Seminar

Erratic behavior of the coefficients of modular forms
Friday, 28 April 2017 - 2:30 pm to 3:00 pm
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SPEAKER: Yuri Bilu (U. de Bordeaux 1)             ABSTRACT: I will speak on a recent joint work with Jean-Marc Deshouillers, Sanoli Gun and Florian Luca. Here is a sample result. Let τ(.) be the classical Ramanujan τ-function defined by
q∏n>0 (1-qn)24 = ∑n>0 τ(n) qn .
The classical work of Rankin implies that both inequalities |τ(n)|<|τ(n+1)| and |τ(n)|>|τ(n+1)| hold for infinitely many n. We generalize this for longer segments of consecutive values of τ.
Let k be a positive integer such that τ(n) is not 0 for n≤ k/2. (This is known to be true for all k < 10<sup>23</sup>, 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 |τ(n+s(1))|<|τ(n+s(2))|<...<|τ(n+s(k))