Waring problem with the Ramanujan $\tau$-function

Abstract

Let $\tau (n)$ be the Ramanujan $\tau$-function. We prove that for any integer $N$ the diophantine equation $$\sum _{i=1}^{74000}\tau (n_i)=N$$ has a solution in positive integers $n_1,n_2,...,n_{74000}$ satisfying the condition $$\underset{1\le i\le 74000}{\max }{n}_i\ll |N{|}^{2/11}+1.$$ We also consider similar questions in the residue ring modulo a large prime $p.$

17 pages

References in corpus (1)

• An invitation to additive prime number theory