A generalised divisor problem (Q2755160)

This memoir is concerned with a generalisation of the classical divisor problems in which one considers NEWLINE\[NEWLINE\Delta_k(x)=\sum_{{n_1,\dots,n_k} \atop {n_1\dots n_k\leq x}} a_{n_1} \dots a_{n_k}-xQ_k(\log x),NEWLINE\]NEWLINE where \(\{a_n\}\) denotes a sequence of complex numbers and \(Q_k(y)\) is an arbitrary polynomial with complex coefficients. We obtain the remarkable conclusion that it is possible to imitate closely the classical case, and under rather general conditions concerning the mean values of the \(a_n\) we show that \(\Delta_k(x)\) cannot be of smaller order than \(x^{\frac{k-1}{2k}}\) as \(x\to\infty\). This improves upon classical results of \textit{H.-E. Richert} [J. Reine Angew. Math. 206, 31-38 (1961; Zbl 0106.03304)] and is essentially best possible.