An additive problem with Ramanujan's function (Q2857874)

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 NEWLINE\[NEWLINE \sum_{i=1}^{7544}\tau(n_i)=N NEWLINE\]NEWLINE has a solution in positive integers \(n_1,n_2,\dots,n_{7544}\) satisfying the condition NEWLINE\[NEWLINE \max_{1\leq i\leq 7544}n_i\ll |N|^{\frac{2}{11}}\,. NEWLINE\]NEWLINE 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), \url{doi:10.3103/S0027132211060106}].