论文标题
在临界线上具有加权Zeta平方测量的多项式矩
Polynomial Moments with a weighted Zeta Square measure on the critical line
论文作者
论文摘要
我们证明了整个关键行$ s = 1/2+it $的矩序列$ \ int t^{2n} |γ（s）ζ|^2dt $的封闭形式的身份。它们是涉及二项式系数，Bernoulli数字，Stirling号码和$π$的有限总和，尤其是由Bettin和Conrey揭示的数字$ζ（n）b_n/n $。他们的主要功率系列身份以及我们以前的工作允许辅助结果的简短证明：计算$ k $ -th衍生词的“指数自动相关”功能的$ 1 $，以\ cite {dh21a}研究。我们还提供了该次要结果的基础和独立的证据。我们作品的起点是Ramanujan在1915年证明的一个了不起的身份。％今天被解释为涉及$γ$和$ζ$函数的Mellin-Plancherel等轴测图。这里研究的时刻顺序，不要与Riemann Zeta函数的矩相混淆，完全表征了临界线上的$ |ζ| $。它们是在Nyman-Beburing标准的一些概括中出现的，但可能对其他各种应用程序以及有关上述数字的众多连接具有独立的兴趣。
We prove closed-form identities for the sequence of moments $\int t^{2n}|Γ(s)ζ(s)|^2dt$ on the whole critical line $s=1/2+it$. They are finite sums involving binomial coefficients, Bernoulli numbers, Stirling numbers and $π$, especially featuring the numbers $ζ(n)B_n/n$ unveiled by Bettin and Conrey. Their main power series identity, together with our previous work, allows for a short proof of an auxiliary result: the computation of the $k$-th derivatives at $1$ of the "exponential auto-correlation" function studied in \cite{DH21a}. We also provide an elementary and self-contained proof of this secondary result. The starting point of our work is a remarkable identity proven by Ramanujan in 1915. %today interpreted as a Mellin-Plancherel isometry involving the $Γ$ and $ζ$ functions. The sequence of moments studied here, not to be confused with the moments of the Riemann zeta function, entirely characterizes $|ζ|$ on the critical line. They arise in some generalizations of the Nyman-Beurling criterion, but might be of independent interest for %various other applications, as well as for the numerous connections concerning the above mentioned numbers.
