Type: Article

The Waring problem with the Ramanujan tau-function. II

M. Z. Garaev; V. C. Garcia; S. V. Konyagin;

Abstract:

Let tau(n) be the Ramanujan tau-function. We prove that for any integer N with vertical bar N vertical bar >= 2 the diophantine equation Sigma(148000)(i=1) tau(n(i)) = N has a solution in positive integers n(1), n(2), ... , n(148000) satisfying the condition max(1 <= i <= 148000) n(i) << vertical bar N vertical bar(2/11)e(-clog vertical bar N vertical bar/log log vertical bar N vertical bar), for some absolute constant c > 0.