Mean Values for a Class of Arithmetic Functions in Short Intervals

TL;DR

This paper develops a general asymptotic formula for certain arithmetic functions in short intervals and explores applications to divisor distribution of square-full numbers and sums of two squares.

Contribution

It introduces a broad asymptotic framework for arithmetic functions in short intervals and applies it to specific number theory problems.

Findings

Asymptotic formulas for arithmetic functions in short intervals

Distribution results for divisors of square-full numbers

Representation of integers as sums of two squares

Abstract

In this paper, we shall establish a rather general asymptotic formula in short intervals for a classe of arithmetic functions and announce two applications about the distribution of divisors of square-full numbers and integers representable as sums of two squares.