This article is cited in 8 papers

Waring's problem with the Ramanujan $\tau$-function

M. Z. Garaev^a, V. C. Garcia^a, S. V. Konyagin<sup>b

a</sup> National Autonomous University of Mexico, Institute of Mathematics
^b M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: We prove that for every integer $N$ the Diophantine equation $\sum_{i=1}^{74000}\tau(n_i)=N$, where $\tau(n)$ is the Ramanujan $\tau$-function, has a solution in positive integers $n_1, n_2,\dots, n_{74000}$ satisfying the condition $\max_{1\le i\le 74000}n_i\,{\ll}|N|^{2/11}+1$. We also consider similar questions in residue fields modulo a large prime $p$.

UDC: 511.34

MSC: 11B83, 11B50, 11P32

Received: 12.07.2006

DOI: 10.4213/im1139

 English version: Izvestiya: Mathematics, 2008, 72:1, 35–46

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