关于对称积分
On the Symmetry Integral
摘要：
我们给出本质有界$f:\N \to \R$的“对称积分”的一级结果，即$I_f(N,h)$，我们得到了一种“平方根抵消”\thinspace界的均方（在$N<x\le 2N$）的“对称”\thinspace，比方说，算术函数$f:=g\ast \1$，其中$g:\N \to \R$是这样的，$\forall ε>0$我们有$g(n)\ll_ε n^ε$，并支持在$[1,Q]$，与$Q\ll N$（因此，$Q$相对于$N$的指数，假设级别$λ:=(\log Q)/(\log N)$是$λ< 1$），其中对称和对$f-$值进行加权，（几乎所有，即除了$o(N)$可能的例外）短间隔$[x-h,x+h]$（在$x$的右/左具有正/负号），对$h$具有轻度限制（例如，$h\to \infty$和$h=o(\sqrt N)$，作为$N\to \infty$）。
英文摘要：
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 $\forall ε>0$ we have $g(n)\ll_ε n^ε$, and supported in $[1,Q]$, with $Q\ll N$ (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, $h\to \infty$ and $h=o(\sqrt N)$, as $N\to \infty$).
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
