论文标题
加权算术总和的振荡
Oscillations in weighted arithmetic sums
论文作者
论文摘要
我们检查了许多涉及$ω（n）$的算术功能的振荡，$ n $的主要因素的总数和$ω（n）$，$ n $的不同主要因素的数量。特别是，我们检查了$s_α（x）= \ sum_ {n \ leq x}（-1）^{n-ω（n）}/n^α$以及$h_α（x）= \ sum_ {n \ leq x}（n \ leq x}（-1）（-1）（-1）^$ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ $ w（x）= \ sum_ {n \ leq x}（-2）^{ω（n）} $。 We show for example that each of the inequalities $S_0(x)<0$, $S_0(x)>3.3\sqrt{x}$, $S_1(x)>0$, and $S_1(x)\sqrt{x}<-3.3$ is true infinitely often, disproving some hypotheses of Sun.
We examine oscillations in a number of sums of arithmetic functions involving $Ω(n)$, the total number of prime factors of $n$, and $ω(n)$, the number of distinct prime factors of $n$. In particular, we examine oscillations in $S_α(x) = \sum_{n\leq x} (-1)^{n - Ω(n)}/n^α$ and in $H_α(x) = \sum_{n\leq x} (-1)^{ω(n)}/n^α$ for $α\in[0,1]$, and in $W(x)=\sum_{n\leq x} (-2)^{Ω(n)}$. We show for example that each of the inequalities $S_0(x)<0$, $S_0(x)>3.3\sqrt{x}$, $S_1(x)>0$, and $S_1(x)\sqrt{x}<-3.3$ is true infinitely often, disproving some hypotheses of Sun.
