The Waring Problem with the Ramanujan $\tau$-Function, II

http://dx.doi.org/10.4153/CMB-2009-022-2
Canad. Math. Bull. 52(2009), 195-199

  Published:2009-06-01
 Printed: Jun 2009

Let $\tau(n)$ be the Ramanujan $\tau$-function. We prove that for any integer $N$ with $|N|\ge 2$ the diophantine equation $$\sum_{i=1}^{148000}\tau(n_i)=N$$ has a solution in positive integers $n_1, n_2,\ldots, n_{148000}$ satisfying the condition $$\max_{1\le i\le 148000}n_i\ll |N|^{2/11}e^{-c\log |N|/\log\log |N|},$$ for some absolute constant $c>0.$