On the Symmetry Integral

TL;DR

This paper establishes a level one bound for the symmetry integral of bounded arithmetic functions, demonstrating a form of square-root cancellation in the mean-square of symmetry sums over short intervals.

Contribution

It provides the first level one result for the symmetry integral of bounded functions, extending understanding of symmetry sums in analytic number theory.

Findings

Achieves a square-root cancellation bound for the symmetry integral.

Validates the bound for functions supported in short intervals with level rac{}{} < 1.

Applicable to almost all short intervals with mild restrictions on h.

Abstract

We give a level one result for the "symmetry integral", say $I_f(N,h)$, of essentially bounded $f:\N \to \R$; i.e., we get a kind of "square-root cancellation" \thinspace bound for the mean-square (in $N<x\le 2N$) of the "symmetry" \thinspace of, say, the arithmetic function $f:=g\ast \1$, where $g:\N \to \R$ is such that $\forall \epsilon>0$ we have $g(n)\ll_{\epsilon} n^{\epsilon}$, and supported in $[1,Q]$, with $Q\ll N$ (so, the exponent of $Q$ relative to $N$, say the level $\lambda:=(\log Q)/(\log N)$ is $\lambda < 1$), where the symmetry sum weights the $f-$values in (almost all, i.e. all but $o(N)$ possible exceptions) the short intervals $[x-h,x+h]$ (with positive/negative sign at the right/left of $x$), with mild restrictions on $h$ (say, $h\to \infty$ and $h=o(\sqrt N)$, as $N\to \infty$).