论文标题
通过广义von Mangoldt功能的Riemann假设
The Riemann Hypothesis via the generalized von Mangoldt Function
论文作者
论文摘要
杰克，格雷厄姆和李最近表明，可以根据von mangoldt函数$λ$的某些扭曲总和的某些渐近估计值来重新制定Riemann假设（RH）。在他们的想法的基础上，对于\ mathbb {n} $中的每个$ k \，我们都会研究\ emph {peneralized von mangoldt函数} $$λ_k（n）：= \ sum_ {d \，\ sum_ {d \，\ mid \ mid \，n}μ（n}μ（d）与RH的类似连接。例如，对于$ k = 2 $，我们表明rh等同于这样的断言，即对于任何固定的$ε> 0 $，估计$$ \ sum_ {n \ leq x}λ_2（n）n^{ - iy} = \ frac = \ frac {2x^{2x^{1-iy} {1- iy} {1- \ log x-c_______________________________________（ - \ frac {2x^{1-iy}}} {（1-iy）^2} +o \ big（x^{1/2}（x +| y |）^ε\ big）$ big）$ big）$均均匀地保持着所有$ x，y \ in \ x，y \ in \ mathbb {r} $ x $ x $ x \ geq 2 $ geq 2 $;因此，RH的有效性受整数中几乎普遍的分布所支配。我们获得了功能的相似结果$$λ^k：= \ mathop {\ usterbrace {\，λ\ star \ cdots \ cdots \starλ\，}} \ limits_ {k \ text {〜copies}}}} \ ,, $ k $ -k $ -fold卷积的Von mangoldt函数。
Gonek, Graham, and Lee have shown recently that the Riemann Hypothesis (RH) can be reformulated in terms of certain asymptotic estimates for twisted sums with von Mangoldt function $Λ$. Building on their ideas, for each $k\in\mathbb{N}$, we study twisted sums with the \emph{generalized von Mangoldt function} $$ Λ_k(n):=\sum_{d\,\mid\,n}μ(d)\Big(\log\frac{n}{d}\,\Big)^k $$ and establish similar connections with RH. For example, for $k=2$ we show that RH is equivalent to the assertion that, for any fixed $ε>0$, the estimate $$ \sum_{n\leq x}Λ_2(n)n^{-iy} =\frac{2x^{1-iy}(\log x-C_0)}{(1-iy)} -\frac{2x^{1-iy}}{(1-iy)^2} +O\big(x^{1/2}(x+|y|)^ε\big) $$ holds uniformly for all $x,y\in\mathbb{R}$, $x\geq 2$; hence, the validity of RH is governed by the distribution of almost-primes in the integers. We obtain similar results for the function $$ Λ^k:=\mathop{\underbrace{\,Λ\star\cdots\starΛ\,}}\limits_{k\text{~copies}}\,, $$ the $k$-fold convolution of the von Mangoldt function.
