On the Modified Selberg Integral

TL;DR

This paper establishes an approximate majorant principle for the modified Selberg integral of bounded functions, providing upper bounds and square-root cancellation for error terms in mean-square estimates over short intervals.

Contribution

It introduces a new upper bound technique for the modified Selberg integral using a related function, achieving square-root cancellation for error terms.

Findings

Provides an upper bound for the modified Selberg integral in terms of a related function.

Achieves square-root cancellation for error terms in mean-square estimates.

Applicable to functions with mild restrictions on the interval length h.

Abstract

We give a kind of \lq \lq approximate majorant principle\rq \rq \thinspace result for the \lq \lq modified Selberg integral\rq \rq, say $\modSel_f(N,h)$, of essentially bounded $f:\N \rightarrow \R$ (i.e., bounded by arbitrary small powers); i.e., we get an upper bound, in terms of the modified Selberg integral of a related function $F$ (with $|f\ast \mu|\ll F\ast \mu$, in the supports intersection), getting a \lq \lq square-root cancellation\rq \rq \thinspace for the error-terms. Here $\modSel_f(N,h)$ is the mean-square (in $N<x\le 2N$) of the \lq \lq averaged short sum\rq \rq \thinspace of, say, $f:=g\ast \1$, minus its expected value; i.e., ${1\over h}\sum_{m\le h}\sum_{0\le |n-x|<m}f(n)-M_f(x,h)$, with expected value $M_f(x,h)$ (say, $\approx h\sum_{d\le x}g(d)/d$); so, this mean-square weights, on average, the $f-$values in (almost all, i.e. all, but $o(N)$ possible exceptions) the short intervals $[x-h,x+h]$, with mild restrictions on \thinspace $h$ \thinspace (say, \thinspace $h\to \infty$ \thinspace and \thinspace $h=o(N)$, when \thinspace $N\to \infty$).