On the Mean Values of the Function $\tau_k(n)$ in Sequences of Natural Numbers

TL;DR

This paper derives an asymptotic formula for the average value of the divisor function _k(n) over specific sequences of natural numbers, extending understanding of its behavior in these contexts.

Contribution

It provides a new asymptotic formula for the mean of _k(n) in special sequences, advancing the analysis of divisor functions in number theory.

Findings

Asymptotic formula for _k(n) mean values in special sequences

Extension of divisor function analysis to new sequences

Enhanced understanding of _k(n) distribution

Abstract

We obtain an asymptotic formula for the mean value of the function $\tau_k(n)$, which is the number of solutions of the equation $x_1...\, x_k=n$ in natural numbers $x_1,..., x_k$ in some special sequences of natural numbers.