The Waring Problem with the Ramanujan τ -Function, II
Canadian mathematical bulletin, Tome 52 (2009) no. 2, pp. 195-199
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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 $c\,>\,0$ .
Garaev, M. Z.; Garcia, V. C.; Konyagin, S. V. The Waring Problem with the Ramanujan τ -Function, II. Canadian mathematical bulletin, Tome 52 (2009) no. 2, pp. 195-199. doi: 10.4153/CMB-2009-022-2
@article{10_4153_CMB_2009_022_2,
author = {Garaev, M. Z. and Garcia, V. C. and Konyagin, S. V.},
title = {The {Waring} {Problem} with the {Ramanujan} \ensuremath{\tau} {-Function,} {II}},
journal = {Canadian mathematical bulletin},
pages = {195--199},
year = {2009},
volume = {52},
number = {2},
doi = {10.4153/CMB-2009-022-2},
url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-2009-022-2/}
}
TY - JOUR AU - Garaev, M. Z. AU - Garcia, V. C. AU - Konyagin, S. V. TI - The Waring Problem with the Ramanujan τ -Function, II JO - Canadian mathematical bulletin PY - 2009 SP - 195 EP - 199 VL - 52 IS - 2 UR - http://geodesic.mathdoc.fr/articles/10.4153/CMB-2009-022-2/ DO - 10.4153/CMB-2009-022-2 ID - 10_4153_CMB_2009_022_2 ER -
%0 Journal Article %A Garaev, M. Z. %A Garcia, V. C. %A Konyagin, S. V. %T The Waring Problem with the Ramanujan τ -Function, II %J Canadian mathematical bulletin %D 2009 %P 195-199 %V 52 %N 2 %U http://geodesic.mathdoc.fr/articles/10.4153/CMB-2009-022-2/ %R 10.4153/CMB-2009-022-2 %F 10_4153_CMB_2009_022_2
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