On higher-power moments of the error term for the divisor problem with congruence conditions
Let, for fixed \(M_1, M_2 \in \mathbb N\), \(d(n;\ell_1,M_1,\ell_2,M_2)\) denote the number of solutions in \(n_1, n_2\) of \(n_1n_2=n, n_i\equiv \ell_i \, ({\roman {mod}} \,M_i) \,(i = 1,2)\). By the divisor problem with congruence conditions one means the estimation of the function \[ \Delta(x;\ell_1,M_1,\ell_2,M_2):= \sum_{n\leq x}d(n;\ell_1,M_1,\ell_2,M_2) \] \[ - \sum_{s_0=0,1}{\roman {Res}}_{s=s_0}\left(\frac{\zeta(s,\lambda_1)\zeta(s,\lambda_2)}{s} \left(\frac{x}{M_1M_2}\right)^s\right). \] Here \(\zeta(s,\lambda)\) denotes the familiar Hurwitz zeta-function with \(\Re s > 1, \lambda_i = \ell_i/M_i\). The author proves that \[ \begin{align*}{& \int_1^T\Delta^3(M_1M_2x;\ell_1,M_1,\ell_2,M_2)dx = C_3\left(\frac{\ell_1}{M_1},\frac{\ell_2}{M_2}\right) T^{7/4} + O(T^{3/2+\varepsilon}),\cr& \int_1^T\Delta^4(M_1M_2x;\ell_1,M_1,\ell_2,M_2)dx = C_4\left(\frac{\ell_1}{M_1},\frac{\ell_2}{M_2}\right) T^{2} + O(T^{53/28+\varepsilon}),\cr}\end{align*} \] Both \(O\)-constants in these formulas depend on \(M_1, M_2\) and \(\varepsilon\), which denotes an arbitrarily small, positive constant, and \(C_3, C_4\) are certain explicit, positive constants. The proof is based on the method which originated with \textit{W. Zhai} [Acta Arith. 114, No. 1, 35--54 (2004; Zbl 1123.11031)].
scientific article; zbMATH DE number 5903711
scientific article