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An additive problem with Ramanujan’s function. (Russian) Zbl 1301.11068

Let \(\tau (n)\) be the be the Ramanujan \(\tau\)-function. Using the circle method, the author proves that for any integer \(N\) the Diophantine equation \[ \sum_{i=1}^{7544}\tau(n_i)=N \] has a solution in positive integers \(n_1,n_2,\dots,n_{7544}\) satisfying the condition \[ \max_{1\leq i\leq 7544}n_i\ll |N|^{\frac{2}{11}}\,. \] This result improves the author’s previous work in [Math. Notes 90, No. 5, 723–729 (2011); translation from Mat. Zametki 90, No. 5, 736–743 (2011; Zbl 1301.11048)] and [“On the representability of integers by the values of the Ramanujan function”, Mosc. Univ. Math. Bull. 66, No. 6, 270–272 (2011); translation from Vestn. Mosk. Univ. Ser. I 2011, No. 6, 49–52 (2011), doi:10.3103/S0027132211060106].

MSC:

11P05 Waring’s problem and variants
11F27 Theta series; Weil representation; theta correspondences
11P55 Applications of the Hardy-Littlewood method

Citations:

Zbl 1301.11048
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Full Text: MNR