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

Published:2009-06-01
Printed: Jun 2009
• M. Z. Garaev
• V. C. Garcia
• S. V. Konyagin
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## Abstract

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 $c>0.$
 MSC Classifications: 11B13 - Additive bases, including sumsets [See also 05B10] 11F30 - Fourier coefficients of automorphic forms