On the Symmetry Integral

2010/07/07 by Giovanni Coppola, Coppola, Giovanni
Mathematics · #Advanced Harmonic Analysis Research #Analytic Number Theory Research #Limits and Structures in Graph Theory #math.NT #msc:11N25 #msc:11N37

paper · pdf · doi:10.48550/arxiv.1007.1018

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arxiv created 2010/07/07 · arxiv updated 2010/07/08

Abstract

We give a level one result for the "symmetry integral", say I_f(N,h), of essentially bounded f:\N → \R; i.e., we get a kind of "square-root cancellation" \thinspace bound for the mean-square (in N<x≤ 2N) of the "symmetry" \thinspace of, say, the arithmetic function f:=g∗ \1, where g:\N → \R is such that ∀ ε>0 we have g(n)≪_ε n^ε, and supported in [1,Q], with Q≪ N (so, the exponent of Q relative to N, say the level λ:=(log Q)/(log N) is λ< 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→ ∞ and h=o(√ N), as N→ ∞).

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