Waring problem with the Ramanujan $\tau$-function

M. Z. Garaev; V. C. Garcia; S. V. Konyagin

Waring problem with the Ramanujan $\tau$-function

Number Theory 2007-05-23 v1

Authors: M. Z. Garaev , V. C. Garcia , S. V. Konyagin

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 $p.$

Keywords

diophantine equations

Cite

@article{arxiv.math/0607169,
  title  = {Waring problem with the Ramanujan $\tau$-function},
  author = {M. Z. Garaev and V. C. Garcia and S. V. Konyagin},
  journal= {arXiv preprint arXiv:math/0607169},
  year   = {2007}
}

Comments

17 pages

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