Stieltjes常数的函数方程
Functional equations for the Stieltjes constants
摘要：
Stieltjes常数$γ_k(a)$表现为Hurwitz zeta函数$ζ(s,a)$关于$s=1$的洛朗展开的规则部分中的系数。我们提出的评价$γ_1(a)$和$γ_2(a)$在合理的论点，感兴趣的理论和计算解析数论和其他地方。我们给出了$γ_0(a)$、$γ_1(a)$和$γ_2(a)$的乘法公式，并指出这些公式是前面提出的加法公式的情况。给出了有理幅角双伽玛函数的积分估计，推广了高斯公式.此外，我们还给出了$γ_k(a)$作为$a \to 0$的渐近形式，以及一种用$\ln(-\ln x)$和有理因子计算被积函数积分的新技术。
英文摘要：
The Stieltjes constants $γ_k(a)$ appear as the coefficients in the regular part of the Laurent expansion of the Hurwitz zeta function $ζ(s,a)$ about $s=1$. We present the evaluation of $γ_1(a)$ and $γ_2(a)$ at rational argument, being of interest to theoretical and computational analytic number theory and elsewhere. We give multiplication formulas for $γ_0(a)$, $γ_1(a)$, and $γ_2(a)$, and point out that these formulas are cases of an addition formula previously presented. We present certain integral evaluations generalizing Gauss' formula for the digamma function at rational argument. In addition, we give the asymptotic form of $γ_k(a)$ as $a \to 0$ as well as a novel technique for evaluating integrals with integrands with $\ln(-\ln x)$ and rational factors.
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