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The Waring Problem with the Ramanujan τ -Function, II

Published online by Cambridge University Press:  20 November 2018

M. Z. Garaev ,
V. C. Garcia  and
S. V. Konyagin
M. Z. Garaev
Affiliation:
Instituto de Matemáticas, Universidad Nacional Autónoma de México, C.P. 58089, Morelia, Michoacán, México e-mail: garaev@matmor.unam.mxgarci@matmor.unam.mx
V. C. Garcia
Affiliation:
Instituto de Matemáticas, Universidad Nacional Autónoma de México, C.P. 58089, Morelia, Michoacán, México e-mail: garaev@matmor.unam.mxgarci@matmor.unam.mx
S. V. Konyagin
Affiliation:
Department of Mechanics and Mathematics, Moscow State University, Moscow, 119992, Russia e-mail: konyagin@ok.ru
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Abstract

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Let $\tau \left( n \right)$ be the Ramanujan $\text{ }\!\!\tau\!\!\text{ }$-function. We prove that for any integer $N$ with $\left| N \right|\,\ge \,2$ the diophantine equation

$$\underset{i=1}{\overset{148000}{\mathop{\sum }}}\,\,\tau \left( {{n}_{i}} \right)\,=\,N$$

has a solution in positive integers ${{n}_{1}},\,{{n}_{2}},\,.\,.\,.\,,\,{{n}_{148000}}$ satisfying the condition

$$\underset{1\le i\le 148000}{\mathop{\max }}\,{{n}_{i}}\ll |N{{|}^{2/11}}{{e}^{-c\log |N|/\log \log |N|}},$$

for some absolute constant $c\,>\,0$.

Keywords

11B1311F30
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 52 , Issue 2 , 01 June 2009 , pp. 195 - 199
Copyright
Copyright © Canadian Mathematical Society 2009

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