具有乘法系数的指数和的有效估计
Effective estimates for exponential sums with multiplicative coefficients
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中文总结 AI 辅助
本文改进了具有乘法系数的指数和估计，去除了Bachman结果中的$\sqrt{\log R}$因子，同时保留Montgomery-Vaughan的原始假设，并证明平方根位移依赖的锐性。
中文摘要 AI 辅助
设$f$为乘法函数，在素数处满足$|f(p)|\le A$，且对所有$x\ge1$有$\sum_{n\le x}|f(n)|^2\le A^2x$。若$|\alpha-a/q|\le q^{-2}$，$(a,q)=1$，且$3\le R\le q\le N/R$，我们证明\\[ \sum_{n\le N}f(n)\operatorname{e}(n\alpha) \ll_A \frac{N}{\log N} +\frac{N}{\sqrt R}\sqrt{\log\log(3R)} \\]其中隐含常数是有效的。Montgomery和Vaughan证明了第二项为$NR^{-1/2}(\log R)^{3/2}$，而对于$1$-有界函数，Bachman将其替换为$NR^{-1/2}\sqrt{\log R\log\log R}$。我们从Bachman的第二项中去掉了因子$\sqrt{\log R}$，同时保留了Montgomery和Vaughan的原始系数假设。一个更精确的估计记录了与有理数的距离。证明结合了短区间上的Brun-Titchmarsh不等式与由Carleson-Hunt定理导出的最大傅里叶估计；局部界允许任意依赖于素数的前缀。我们还证明了平方根位移依赖性的锐性。
英文摘要
Let $f$ be multiplicative, with $|f(p)|\le A$ at primes and $\sum_{n\le x}|f(n)|^2\le A^2x$ for every $x\ge1$. If $|α-a/q|\le q^{-2}$, $(a,q)=1$, and $3\le R\le q\le N/R$, we prove \[ \sum_{n\le N}f(n)\operatorname{e}(nα) \ll_A \frac{N}{\log N} +\frac{N}{\sqrt R}\sqrt{\log\log(3R)} \] with effective implied constants. Montgomery and Vaughan proved this with second term $NR^{-1/2}(\log R)^{3/2}$, and, for $1$-bounded functions, Bachman replaced it by $NR^{-1/2}\sqrt{\log R\log\log R}$. We remove the factor $\sqrt{\log R}$ from Bachman's second term while retaining the original coefficient hypotheses of Montgomery and Vaughan. A more precise estimate records the distance from a rational number. The proof combines the Brun-Titchmarsh inequality on short intervals with maximal Fourier estimates derived from the Carleson-Hunt theorem; the local bounds permit arbitrary prime-dependent prefixes. We also prove sharpness of the square-root displacement dependence.
发表机构
- RAND Corporation（兰德公司）
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