On incommensurability of two-dimensional standard integer lattice and a lattice generated by vectors, which have zeta values as coordinates

Special zeta values using tensor powers of Drinfeld modules

We study tensor powers of rank 1 sign-normalized Drinfeld A-modules, where A is the coordinate ring of an elliptic curve over a finite field. Using the theory of vector valued Anderson generating functions, we give formulas for the coefficients of the logarithm and exponential functions associated to these A-modules. We then show that there exists a vector whose bottom coordinate contains a Goss zeta value, whose evaluation under the exponential function is defined over the Hilbert class field. This allows us to prove the transcendence of Goss zeta values and periods of Drinfeld modules as wel…

Deformation of Multiple Zeta Values and Their Logarithmic Interpretation in Positive Characteristic

Pellarin introduced the deformation of multiple zeta values of Thakur as elements over Tate algebras. In this paper, we relate these values to a certain coordinate of a higher dimensional Drinfeld module over Tate algebras which we will introduce. Moreover, we define multiple polylogarithms in our setting and represent deformation of multiple zeta values as a linear combination of multiple polylogarithms.

Taylor coefficients of Anderson-Thakur series and explicit formulae

For each positive characteristic multiple zeta value (defined by Thakur), the first and third authors constructed a $t$-module together with an algebraic point such that a specified coordinate of the logarithmic vector of the algebraic point is a rational multiple of that multiple zeta value. The objective of this paper is to use the Taylor coefficients of Anderson-Thakur series and $t$-motivic Carlitz multiple star polylogarithms to give explicit formulae for all of the coordinates of this logarithmic vector.

Rogers-Ramanujan连分数的对数矩与奇数zeta值

Logarithmic moments of the Rogers-Ramanujan continued fraction and odd zeta values

该研究探讨Rogers-Ramanujan连分数的尖点展开对其普通与对数Mellin矩的控制作用，证明相关系数的代数性质，确定归一化矩生成函数的极点，并建立反射恒等式等结果。

$k$-free lattice points in random walks

Let $\mathbb{Z}^2$ be the two-dimensional integer lattice. For an integer $k\geq 1$, a non-zero lattice point is $k$-free if the greatest common divisor of its coordinates is a $k$-free number. We consider the proportions of $k$-free and twin $k$-free lattice points on a path of an $α$-random walker in $\mathbb{Z}^2$. Using the second-moment method and tools from analytic number theory, we prove that these two proportions are $1/ζ(2k)$ and $\prod_{p}(1-2p^{-2k})$, respectively, where $ζ$ is the Riemann zeta function and the infinite product takes over all primes.

$\boldsymbol{\text{PGL}}_d$ 的标准算术商的顶点体积、格极小值尾和高度泽塔函数

Vertex volumes, lattice-minima tails, and height zeta functions for the standard arithmetic quotient of $\operatorname{PGL}_d$

该研究针对 $\text{PGL}_d$ 的标准算术商，基于建筑理论计算顶点体积，引入同伦不变归一性格极小值高度，确定其可积阈值与尖点尾估计，推导高度泽塔函数的收敛域与解析延拓性质，明确计算了 $d=3,4,5$ 时的有理函数。