Third power moments of the error terms corresponding to certain arithmetic functions
Let \(d(n)\) be the number of divisors of \(n\) and \[ \sum_{n \leq x} d(n) = x \log x + (2C - 1) + \Delta (x), \] where \(C\) is Euler's constant, and \(x\) is a large variable. \textit{K. M. Tsang} [Proc. Lond. Math. Soc. (3) 65, No. 1, 65--84 (1992; Zbl 0725.11046)] proved for the remainder \(\Delta (x)\) the very interesting result \[ \int^T_1 \bigl( \Delta (x) \bigr)^3 \,dx = a_1 T^{7/4} + O(T^{47/28 + \varepsilon}) \] with a well-defined positive constant \(a_1\). The authors consider the divisor function \[ d(l_1, M_1; l_2, M_2; n) = \# \bigl\{ (n_1 n_2) \in \mathbb N^2 : n_1 n_2 = n, \;n_i \equiv l_i \pmod {M_i} \bigr\}. \] Analogously, let \(\Delta (l_1, M_1; l_2, M_2; n)\) be the remainder in the asymptotic expansion of the sum \(\sum_{n \leq x} d(l_1, M_1; l_2, M_2; n)\) and \(E(y) = \Delta (l_1, M_1, l_2, M_2; M_1 M_2y)\). Then it is proved that the remainder has the same behaviour as before, namely \[ \int^T_1 \bigl( E(y) \bigr)^3 \,dy = b_1 T^{7/4} + O(T^{47/28 + \varepsilon} \bigr), \] where again \(b_1\) is a well-defined positive constant. The asymptotic expansion holds uniformly in \(1 \leq l_1 \leq M_1\), \(1 \leq l_2 \leq M_2\).
asymptotic results on arithmetic functions
scientific article; zbMATH DE number 824172
scientific article