论文标题
格雷厄姆对芭芭·弗洛夫问题的估计的概括
A Generalization of Graham's Estimate on the Barban-Vehov Problem
论文作者
论文摘要
假设$ \ {λ_d\} $是Selberg的筛子权重，$ 1 \ le W <y \ le x $。 Graham对Barban-Vehov问题的估计表明，$ \ sum_ {1 \ le n \ le x}（\ sum_ {d | n}λ_d）^2 = \ frac {x} {x} {\ log（y/w）} + O（我们证明了这一估算值的类似物，以对任意数字字段$ k $的理想进行总和。我们的渐近估计保持不变。唯一的区别是，有效的错误项可能取决于$ k $的算术。我们的创新涉及对理想而不是整数的多次计数结果。值得注意的是，某些结果是非平凡的概括。此外，我们证明了一种推论，可导致新的零密度估计值。
Suppose $\{ λ_d\}$ are Selberg's sieve weights and $1 \le w < y \le x$. Graham's estimate on the Barban-Vehov problem shows that $\sum_{1 \le n \le x} (\sum_{d|n} λ_d)^2 = \frac{x}{\log(y/w)} + O(\frac{x}{\log^2(y/w)})$. We prove an analogue of this estimate for a sum over ideals of an arbitrary number field $k$. Our asymptotic estimate remains the same; the only difference is that the effective error term may depend on arithmetics of $k$. Our innovation involves multiple counting results on ideals instead of integers. Notably, some of the results are nontrivial generalizations. Furthermore, we prove a corollary that leads to a new zero density estimate.
