Type: Article

The Waring problem with Ramanujan's tau-function

M. Z. Garaev; V. K. Garsiya; S. V. Konyagin;

Abstract:

We prove that for every integer N the Diophantine equation Sigma(74000)(i=1) tau(n(i)) = N, where tau(n) is the Ramanujan tau-function, has a solution in positive integers n(1),n(2),..,n(74000) satisfying the condition max(1 <= i <= 74000) n(i) << |N|(2/11)+1. We also consider similar questions in residue fields modulo a large prime p.