本文旨在拓展Kurokawa与Wakayama在2003年开展的研究，重点聚焦Hurwitz zeta函数的$q$-模拟。我们具体研究了$q$-模拟Hurwitz zeta函数在$s=1$附近洛朗级数展开的系数，推导得到了$q$-Hurwitz zeta函数洛朗级数前两项系数的闭式表达式。此外，利用双伽马函数的反射公式与伯努利多项式恒等式，我们探究了当$q>1$且$0 < x <1$时，与$γ_0(q,x)$相关的超越性结果，其中$γ_0(q,x)$是$q$-Hurwitz zeta函数在$s=1$附近洛朗级数展开中的常数项。\n 进一步地，我们提出了一个猜想：在互素条件下，$γ_0(q,x)$在有理自变量处的特殊值与1在有理数域上线性无关。最后，我们证明了这些数中至少有超过一半的数在有理数域上线性无关。
英文摘要：
In this article, our aim is to extend the research conducted by Kurokawa and Wakayama in 2003, particularly focusing on the $q$-analogue of the Hurwitz zeta function. Our specific emphasis lies in exploring the coefficients in the Laurent series expansion of a $q$-analogue of the Hurwitz zeta function around $s=1$. We establish the closed-form expressions for the first two coefficients in the Laurent series of the $q$-Hurwitz zeta function. Additionally, utilizing the reflection formula for the digamma function and the identity of Bernoulli polynomials, we explore transcendence results related to $γ_0(q,x)$ for $q>1$ and $0 < x <1$, where $γ_0(q,x)$ is the constant term which appears in the Laurent series expansion of $q$-Hurwitz zeta function around $s=1$.
Furthermore, we put forth a conjecture about the linear independence of special values of $γ_0(q,x)$ along with $1$ at rational arguments with co-prime conditions, over the field of rational numbers. Finally, we show that at least one more than half of the numbers are linearly independent over the field of rationals.
The connection between derivatives of $L(s,f)$ for periodic arithmetical functions $f$ at $s=1$ and generalized Stieltjes constants has been noted earlier. In this paper, we utilize this link to throw light on the arithmetic nature of $L'(1,f)$ and certain Stieltjes constants. In particular, if $p$ is an odd prime greater than $7$, then we deduce the transcendence of at least $(p-7)/2$ of the generalized Stieltjes constants, $\{ γ_1(a,p) : 1 \leq a < p \}$, conditional on a conjecture of S. Gun, M. R. Murty and P. Rath.
Stieltjes constants for Hurwitz-type L-series attached to periodic arithmetic functions
Laboratoire de Mathématiques et Modélisation d’Évry（埃夫里数学与建模实验室）；National Higher School of Mathematics, Scientific and Technology Hub of Sidi Abdellah（西迪阿卜杜拉科学和技术枢纽国立高等数学学院）
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