arXivDaily arXiv每日学术速递 周一至周五更新
arXiv 1206.0255math.NT
收藏

Hardy-Littlewood数的Cesàro平均

A Cesàro Average of Hardy-Littlewood numbers

Alessandro Languasco, Alessandro Zaccagnini

备注：
submitted
发表信息：
J. Math. Anal. Appl. 401 (2013), no. 2, 568--577
阅读 PDF
arXiv 页面 ↗HTML 原文 ↗

摘要：

设$Λ$为von Mangoldt函数，$r_{\textit{HL}}(n) = \sum_{m_1 + m_2^2 = n} Λ(m_1),$为Hardy-Littlewood数的计数函数。设$N$是一个足够大的整数。我们证明了$k > 1$的$$\begin{align}\sum_{n \le N} r_{\textit{HL}}(n) \frac{(1 - n/N)^k}{Γ(k + 1)} &= \frac{π^{1 / 2}}2 \frac{N^{3 / 2}}{Γ(k + 5 / 2)} - \frac 12 \frac{N}{Γ(k + 2)} - \frac{π^{1 / 2}}2 \sum_ρ \frac{Γ(ρ)}{Γ(k + 3 / 2 + ρ)} N^{1 / 2 + ρ}\\ &+ 1/2 \sum_ρ \frac{Γ(ρ)}{Γ(k + 1 + ρ)} N^ρ + \frac{N^{3 / 4 - k / 2}}{π^{k + 1}} \sum_{\ell \ge 1} \frac{J_{k + 3 / 2} (2 π\ell N^{1 / 2})}{\ell^{k + 3 / 2}}\\ &- \frac{N^{1 / 4 - k / 2}}{π^k} \sum_ρ Γ(ρ) \frac{N^{ρ/ 2}}{π^ρ} \sum_{\ell \ge 1} \frac{J_{k + 1 / 2 + ρ} (2 π\ell N^{1 / 2})} {\ell^{k + 1 / 2 + ρ}} + \mathcal{O}_k(1).\end{align}$$，其中$ρ$遍历Riemann zeta函数$ζ(s)$的非平凡零点，$J_ν (u)$表示复阶$ν$和实宗量$u$的贝塞尔函数.

英文摘要：

Let $Λ$ be the von Mangoldt function and $r_{\textit{HL}}(n) = \sum_{m_1 + m_2^2 = n} Λ(m_1),$ be the counting function for the Hardy-Littlewood numbers. Let $N$ be a sufficiently large integer. We prove that $$\begin{align}\sum_{n \le N} r_{\textit{HL}}(n) \frac{(1 - n/N)^k}{Γ(k + 1)} &= \frac{π^{1 / 2}}2 \frac{N^{3 / 2}}{Γ(k + 5 / 2)} - \frac 12 \frac{N}{Γ(k + 2)} - \frac{π^{1 / 2}}2 \sum_ρ \frac{Γ(ρ)}{Γ(k + 3 / 2 + ρ)} N^{1 / 2 + ρ}\\ &+ 1/2 \sum_ρ \frac{Γ(ρ)}{Γ(k + 1 + ρ)} N^ρ + \frac{N^{3 / 4 - k / 2}}{π^{k + 1}} \sum_{\ell \ge 1} \frac{J_{k + 3 / 2} (2 π\ell N^{1 / 2})}{\ell^{k + 3 / 2}}\\ &- \frac{N^{1 / 4 - k / 2}}{π^k} \sum_ρ Γ(ρ) \frac{N^{ρ/ 2}}{π^ρ} \sum_{\ell \ge 1} \frac{J_{k + 1 / 2 + ρ} (2 π\ell N^{1 / 2})} {\ell^{k + 1 / 2 + ρ}} + \mathcal{O}_k(1).\end{align}$$ for $k > 1$, where $ρ$ runs over the non-trivial zeros of the Riemann zeta-function $ζ(s)$ and $J_ν (u)$ denotes the Bessel function of complex order $ν$ and real argument $u$.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

↑