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 $$ \max_{1\le i\le 74000}n_i\ll |N|^{2/11}+1. $$ We also consider similar questions in the residue ring modulo a large prime $p.$
- Publication:
-
arXiv Mathematics e-prints
- Pub Date:
- July 2006
- DOI:
- 10.48550/arXiv.math/0607169
- arXiv:
- arXiv:math/0607169
- Bibcode:
- 2006math......7169G
- Keywords:
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- Mathematics - Number Theory;
- 11B13;
- 11F35
- E-Print:
- 17 pages