Some optimal links between generations of correlation averages

TL;DR

This paper explores optimal relationships between different correlation averages of bounded arithmetic functions, providing bounds that connect sums over shifts, short intervals, and weighted sums, with implications for understanding their collective behavior.

Contribution

It establishes the most effective links between bounds for various correlation sums of bounded arithmetic functions, advancing the theoretical understanding of their interrelations.

Findings

Derived optimal bounds connecting different correlation sums

Established relationships between sums over shifts and short intervals

Provided a framework for analyzing correlation averages in number theory

Abstract

For a real-valued and essentially bounded arithmetic function $f$, i.e., $f(n)\ll_{\varepsilon}\!n^{\varepsilon},\,\forall\varepsilon\!>\!0$, we \enspace give some optimal links between non-trivial bounds for the sums $\sum_{h\le H}\sum_{N<n\le 2N}f(n)f(n-h)$, $\sum_{N<x\le 2N} \big| \sum_{x<n\le x+H}f(n)\big|^2$ and $\sum_{N<n\le 2N} \big| \sum_{0\le |n-x|\le H}\big(1-{|n-x|\over H}\big)f(n)\big|^2$, with $H=o(N)$ as $N\to\infty$.