脆数上的广义除数函数均值问题

The Mean Value Problem of Generalized Divisor Functions over Friable Numbers

摘要: 设$P\left(n\right)$ 为正整数$n$ 的最大素因子，并约定$P\left(1\right)=1$ 。本文研究单侧脆数条件下的广义不对称除数求和$D\left(a,b;x,y\right)={\displaystyle \sum _{n^a{m}^b\le x,P\left(n\right)\le y}1}$ ，其中$a,b$ 为互不相等的固定正整数。在$\exp \left\{{\left(\log \log x\right)}^{5/3+\epsilon }\right\}\le y\le x$ 的范围内，利用Hildebrand的脆数分布一致估计、Dirichlet双曲求和和Stieltjes分部积分，得到关于$y$ 一致的渐近公式。对$a>b$ ，保留截断Euler乘积给出的首项及含$\zeta \left(b/a\right)$ 的次主项。对$a<b$ ，通过绝对收敛的$m$ -和得到单一主项及误差。两种情形的差异是因为脆数限制只施加在变量$n$ 上。

Abstract: Let $P\left(n\right)$ denote the largest prime factor of a positive integer $n$ , with the convention $P\left(1\right)=1$ . We investigate the asymmetric generalized divisor sum under a one-sided friability constraint, $D\left(a,b;x,y\right)={\displaystyle \sum _{n^a{m}^b\le x,P\left(n\right)\le y}1}$ , where $a$ and $b$ are distinct fixed positive integers. Uniformly in the range $\exp \left\{{\left(\log \log x\right)}^{5/3+\epsilon }\right\}\le y\le x$ , we derive asymptotic formulae uniform in $y$ by combining Hildebrand’s uniform estimate for the distribution of friable integers with Dirichlet’s hyperbola method and Stieltjes partial summation. When $a>b$ , the resulting expansion contains a leading term involving a truncated Euler product and a secondary term involving $\zeta \left(b/a\right)$ . When $a<b$ , the absolute convergence of the $m$ -sum yields a single main term together with an error term. The difference between these two cases arises from the fact that the friability constraint is imposed only on the variable $n$ .