On the Symmetry Integral

Abstract

We give a level one result for the "symmetry integral", say $I_f(N,h)$, of essentially bounded $f:\mathbb{N}\to \mathbb{R}$; 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 $g:\mathbb{N}\to \mathbb{R}$ 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 $N\to \infty$).

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References in corpus (1)

• On the symmetry of arithmetical functions in almost all short intervals,IV

Cited by in corpus (1)

• On the symmetry of primes