Double integrals and infinite products for some classical constants via analytic continuations of Lerch's transcendent

Multiple Lerch zeta functions and an idea of Ramanujan

In this article, we derive meromorphic continuation of multiple Lerch zeta functions by generalising an elegant identity of Ramanujan. Further, we describe the set of all possible singularities of these functions. Finally, for the multiple Hurwitz zeta functions, we list the exact set of singularities.

Generalizations of Guillera-Sondow's double integral formulas

We evaluate certain multidimensional integrals in terms of the Lerch transcendent function $Φ$, generalizing Guillera-Sondow's formulas. As an application, we get new representations of classical constants like Euler's constant $γ$ and $\ln(4/π)$.

The Lerch zeta function III. Polylogarithms and special values

This paper studies algebraic and analytic structures associated with the Lerch zeta function, extending the complex variables viewpoint taken in part II. The Lerch transcendent $Φ(s, z, c)$ is obtained from the Lerch zeta function $ζ(s, a, c)$ by the change of variable $z=e^{2 πi a}$. We show that it analytically continues to a maximal domain of holomorphy in three complex variables $(s, z, c)$ as a multivalued function defined over the base manifold ${\bf C} \times P^1({\bf C} \smallsetminus \{0, 1, \infty\}) \times ({\bf C}\smallsetminus {\bf Z})$. and compute the monodromy functions definin…

Lerch的$Φ$与正整数的多对数函数

Lerch's $Φ$ and the Polylogarithm at the Positive Integers

本文回顾了与$\HP_k(n)$相关的部分傅里叶和的闭形式，并建立了$\HP(n)$的渐近表达式，从而得到完整傅里叶级数的公式，进而推导出Lerch超越函数$Φ(e^z,k,b)$和多对数函数$\mathrm{Li}_{k}(e^{z})$在正整数$k$处的公式。

Lerch zeta函数在最大全纯域上的显式表达式

An explicit expression of the Lerch zeta function on maximal domains of holomorphy

该研究给出Lerch zeta函数的显式表达式，实现其多变量到最大全纯域的解析延拓并给出特殊值扩展公式，同时证明Lerch与Apostol函数方程本质一致。

Logarithmic Integrals: A Review from Gradshteyn and Ryzhik to Recent Times

The need to evaluate Logarithmic integrals is ubiquitous in essentially all quantitative areas including mathematical sciences, physical sciences. Some recent developments in Physics namely Feynman diagrams deals with the evaluation of complicated integrals involving logarithmic functions. This work deals with a systematic review of logarithmic integrals starting from Malmsten integrals to classical collection of Integrals, Series and Products by I. S. Gradshteyn and I. M. Ryzhik [1] to recent times. The evaluation of these types of integrals involves higher transcendental functions (i.e., Hur…

含三个变量的黎曼函数方程的一个推广

A Generalization of the Riemann Functional Equation with Three Variables

该研究利用赫尔维茨ζ函数与勒让德Φ函数的对称性，将黎曼函数方程推广至含三个变量的形式，扩展其定义域至所有实数b，同时解决了勒让德Φ函数有限求和的解析延拓问题。

Zeta Series Generating Function Transformations Related to Generalized Stirling Numbers and Partial Sums of the Hurwitz Zeta Function

We define a generalized class of modified zeta series transformations generating the partial sums of the Hurwitz zeta function and series expansions of the Lerch transcendent function. The new transformation coefficients we define within the article satisfy expansions by generalized harmonic number sequences, or the partial sums of the Hurwitz zeta function, which are analogous to known properties for the Stirling numbers of the first kind and for the known transformation coefficients employed to enumerate variants of the polylogarithm function series. Applications of the new results we prove …

适用于整个复平面的ζ函数及其推广形式的最简积分

Simplest Integrals for the Zeta Function and its Generalizations Valid in All $\mathbb{C}$

本文扩展Faulhaber公式得到黎曼ζ函数公式，进而推导其推广形式的公式，所用方法涉及特定Lerch transcendent组合的极限计算，相关结果可用于求H_{-k}(n)的泰勒级数。

解析数论中的阿佩尔序列：移位展开与狄利克雷L-函数

Appell Sequences in Analytic Number Theory: Shifted Expansions and Dirichlet L-Functions

该研究将阿佩尔移位定理置于博雷尔-拉普拉斯框架，扩展阿佩尔-梅林框架至狄利克雷L-函数，构造生成函数并确立多项式R_n^χ的五项结构性质，相关结果为首次报道。

Bohr的最后一问题在整体性假设下：一种调查与初步简化

Bohr's Last Problem Under the Entirety Hypothesis: A Survey with Initial Reductions

本文探讨了在整体性假设下Dirichlet级数的Lindelöf假设，分析了通过 Mellin 积分变形将μ_L转化为残差和的过程，并讨论了Kahane的例子及Cantor狄利克雷级数的构造。

Coaction and double-copy properties of configuration-space integrals at genus zero

We investigate configuration-space integrals over punctured Riemann spheres from the viewpoint of the motivic Galois coaction and double-copy structures generalizing the Kawai-Lewellen-Tye (KLT) relations in string theory. For this purpose, explicit bases of twisted cycles and cocycles are worked out whose orthonormality simplifies the coaction. We present methods to efficiently perform and organize the expansions of configuration-space integrals in the inverse string tension $α'$ or the dimensional-regularization parameter $ε$ of Feynman integrals. Generating-function techniques open up a new…

Mizuno 乘积公式的交替推广与修正 gamma 函数

An alternating Mizuno formula, modified gamma functions, and double sine special values

本文建立了 Mizuno 乘积公式的完整交替推广，引入修正 gamma 函数的交替版本，并推导出 Wallis 乘积、Kurokawa--Wakayama 型公式及多重交替 gamma 函数的性质，最后给出 Shintani 双正弦函数交替模拟的算术二分法。