Geodesic
Waring's problem with the Ramanujan $\tau$-function
Izvestiya. Mathematics , Tome 72 (2008) no. 1, pp. 35-46

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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 $p$. 
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     author = {M. Z. Garaev and V. C. Garcia and S. V. Konyagin},
     title = {Waring's problem with the {Ramanujan} $\tau$-function},
     journal = {Izvestiya. Mathematics },
     pages = {35--46},
     publisher = {mathdoc},
     volume = {72},
     number = {1},
     year = {2008},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_2008_72_1_a1/}
}
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M. Z. Garaev; V. C. Garcia; S. V. Konyagin. Waring's problem with the Ramanujan $\tau$-function. Izvestiya. Mathematics , Tome 72 (2008) no. 1, pp. 35-46. http://geodesic.mathdoc.fr/item/IM2_2008_72_1_a1/