标题： 一个关于更大更光滑函数的刘定理 显示英文标题

标题： A Shiu Theorem for Larger and Smoother Functions

摘要： 在本文中，我们将Shiu的Brun-Titchmarsh定理推广，以允许更大的和/或光滑支撑的函数。 特别是，令$f$为一个非负乘法函数。 我们证明，如果存在一个$\beta<1$使得$f(p^l)\ll (\log\log x)^{l\beta}$对每个素数$p$和每个$l>1$成立，并且如果$f(n)\ll \max\{n^\epsilon,(\log x)^\epsilon\}$对每个$\epsilon>0$成立，那么$$\sum_{\substack{x\leq n\leq x+y \\ n\equiv a\pmod k}}f(n)\ll \frac{x}{\phi(k)(\log x)^{1-\epsilon_0}}\exp\left(\sum_{\substack{p\leq x \\ p\nmid k}}\frac{f(p)}{p}\right)$$对每个$\epsilon_0>0$成立，其中$x$、$y$和$k$如同在Shiu的原始论文中所定义，而$(a,k)=1$。 此外，我们证明了如果 $f$是一个 $Q$-光滑支撑函数，那么存在一个常数 $C$，使得 $$\sum_{\substack{x\leq n\leq x+y \\ n\equiv a\pmod k}}f(n)\ll \frac{x}{\phi(k)(\log x)^{1-\epsilon_0}}\exp\left(\sum_{\substack{p\leq x \\ p\nmid k}}\frac{f(p)}{p}\right)\rho(u)^C,$$ 其中 $u=\frac{\log x}{\log Q}$，$\rho$是 Dickman-de Bruijn 函数，而 $C$取决于我们选择 $f(p^l)\leq A_1^l$或 $f(p^l)\ll (\log\log x)^{l\beta}$的界限。 我们还给出了对除数函数到大幂次以及短区间内光滑数的应用。 显示英文摘要

摘要： In this paper, we broaden Shiu's Brun-Titchmarsh theorem to allow for functions that are larger and/or smooth-supported. In particular, let $f$ be a nonnegative multiplicative function. We prove that if there exists a $\beta<1$ such that $f(p^l)\ll (\log\log x)^{l\beta}$ for every prime $p$ and every $l>1$, and if $f(n)\ll \max\{n^\epsilon,(\log x)^\epsilon\}$ for every $\epsilon>0$, then $$\sum_{\substack{x\leq n\leq x+y \\ n\equiv a\pmod k}}f(n)\ll \frac{x}{\phi(k)(\log x)^{1-\epsilon_0}}\exp\left(\sum_{\substack{p\leq x \\ p\nmid k}}\frac{f(p)}{p}\right)$$ for every $\epsilon_0>0$, where $x$, $y$, and $k$ are as they were in Shiu's original paper and $(a,k)=1$. Moreover, we prove that if $f$ is a $Q$-smooth-supported function then there exists a constant $C$ for which $$\sum_{\substack{x\leq n\leq x+y \\ n\equiv a\pmod k}}f(n)\ll \frac{x}{\phi(k)(\log x)^{1-\epsilon_0}}\exp\left(\sum_{\substack{p\leq x \\ p\nmid k}}\frac{f(p)}{p}\right)\rho(u)^C,$$ where $u=\frac{\log x}{\log Q}$, $\rho$ is the Dickman-de Bruijn function, and $C$ depends on whether we choose the bound of $f(p^l)\leq A_1^l$ or $f(p^l)\ll (\log\log x)^{l\beta}$. We also give applications to both the divisor function to large powers and to smooth numbers in short intervals.