AI 中文总结
本文引入整数集上的离散双曲正割分布族，利用Ramanujan级数和椭圆函数构造缩放版本，计算矩并得到闭式级数值，最后应用于围道积分评估。
AI 中文摘要
我们在$\mathbb{Z}$上引入一族离散双曲正割分布，其归一化常数源于Ramanujan计算的级数值，并以Gauß常数$G=\varpi/\pi$表示，其中$\varpi=\Gamma^2(1/4)/(2\sqrt{2\pi})$为lemniscate常数。利用椭圆lambda-star函数$\lambda^*$，我们构造了这些分布的缩放版本，对于$r \in \mathbb{N}$，以$\sqrt{r}$和$1/\sqrt{r}$为参数。对于第一个这样的分布，我们通过Poisson求和以闭式形式计算了直到八阶的矩，利用了双曲正割是$-2\pi i$-Fourier变换的不动点这一事实。作为副产品，我们获得了双曲正割奇次幂级数直到九阶的闭式值，例如$\sum_{k \in \mathbb{Z}} \operatorname{sech}^3(\pi k) = (G^3+G)/\sqrt{2}$，并从概率论角度重新解释了若干经典Ramanujan级数。最后，我们将结果应用于评估临界线上Dirichlet beta函数、gamma函数和Riemann zeta函数乘积的围道积分。
英文摘要
We introduce a family of discrete hyperbolic secant distributions on $\mathbb{Z}$, whose normalizing constants arise from series values calculated by Ramanujan and are expressed in terms of Gauß' constant $G=\varpi/π$, where $\varpi=Γ^2(1/4)/(2\sqrt{2π})$ is the lemniscate constant. Using the elliptic lambda-star function $λ^*$, we construct scaled versions of these distributions parametrized by $\sqrt{r}$ and $1/\sqrt{r}$ for $r \in \mathbb{N}$. For the first such distribution, we compute the moments up to the eighth degree in closed form via Poisson summation, exploiting that the hyperbolic secant is a fixed point of the $-2πi$-Fourier transform. As a byproduct, we obtain closed-form values for series of odd powers of the hyperbolic secant up to the ninth degree, e.\,g.\ $\sum_{k \in \mathbb{Z}} \operatorname{sech}^3(πk) = (G^3+G)/\sqrt{2}$, and reinterpret several classical Ramanujan series probabilistically. Finally, we apply our results to evaluate a contour integral, on the critical line, of the product of Dirichlet's beta, gamma, and Riemann zeta functions.
