An explicit result for short sums of positive arithmetic functions

TL;DR

This paper establishes explicit bounds for short sums of specific non-negative arithmetic functions and applies these bounds to derive concrete estimates for the Erdős-Hooley Δ-function in short intervals.

Contribution

It provides the first explicit bounds for short sums of certain positive arithmetic functions and applies them to improve estimates of the Erdős-Hooley Δ-function.

Findings

Explicit bounds for short sums of arithmetic functions

Two new explicit estimates for the Erdős-Hooley Δ-function

Enhanced understanding of the distribution of arithmetic functions in short intervals

Abstract

We prove a totally explicit bound for short sums of certain non-negative arithmetic functions satisfying a general growth condition, and apply this result to derive two explicit estimates for the Erd\H{o}s-Hooley $\Delta$-function in short intervals.