发表机构
Westlake University; CNRS, IMJ-PRG, Université Paris Cité（西湖大学; 法国国家科学研究中心，IMJ-PRG，巴黎西岱大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究证明广义斯特林-拉马努金常数等相关数论量为指数周期，推导其积分表达式，还将结果推广到广义扭曲情形并关联L函数导数，为指数周期研究提供新的实例与理论拓展。
AI 中文摘要
对于n≥0，斯特林-拉马努金常数Sₙ是发散级数∑_{k≥1}kⁿlogk的拉马努金求和，这些常数是基域ℚ(t,e⁻ᵗ)上的指数周期。我们将该结果推广到一类广泛的广义斯特林-拉马努金常数：给定多项式P和Q，满足当ℜs>0时ℜQ(s)>0，常数S(P,Q)是级数∑_{k≥1}P(k)logQ(k)的拉马努金求和，它们是基域𝕂ₚ((ω))(t,e⁻ᵗ,(e⁻ᵠᵗ))上的指数周期，其中𝕂ₚ是P的定义域，(ω)是Q的零点。这些常数与Hurwitz ζ函数在负整数处的导数ζ'_H(-n,w)相关，我们证明后者是基域ℚ(t,e⁻ᵗ,w,e⁻ʷᵗ)上的指数周期；它们可通过Bendersky Γ函数Ĝₙ表示，且当ℜw>0时logĜₙ(w)是基域ℚ(t,e⁻ᵗ,w,e⁻ʷᵗ)上的指数周期。球面上拉普拉斯算子的ζ正则化行列式的对数是基域ℚ(t,e⁻ᵗ)上的指数周期，值ζ(2n+1)/π²ⁿ是基域ℚ(t,e⁻ᵗ)上的指数周期。对于周期函数χ:ℤ→ℂ，我们将拉马努金求和推广到扭曲级数∑_{k≥1}χ(k)P(k)logQ(k)，定义广义扭曲斯特林-拉马努金常数S_χ(P,Q)，我们推导积分公式证明它们是ℚ(χ)(t,e⁻ᵗ)上的指数周期；对于n≥0，L函数的导数L'_χ(-n)可由这些常数表示，且是基域ℚ(χ)(t,e⁻ᵗ)上的指数周期。
英文摘要
For $n\geq 0$, Stirling-Ramanujan constants $S_n$ are the Ramanujan summation of the divergent series $\sum_{k\geq 1} k^n\log k$. These constants are exponential periods over the exponential base field $\mathbb{E}=\mathbb{Q}(t,e^{-t})$. We generalize this result to a broad class of General Stirling-Ramanujan constants. Given polynomials $P$ and $Q$, with $\Re Q(s)>0$ for $\Re s >0$, the constant $S(P,Q)$ is the Ramanujan summation of the series $\sum_{k\geq 1} P(k)\log Q(k)$. They are exponential periods over $\mathbb{K}_P((ω)) (t,e^{-t}, (e^{-ωt}))$ where $\mathbb{K}_P$ is the field of definition of $P$ and $(ω)$ are the zeros of $Q$. These constants are related to the derivatives $ζ'_H(-n,w)$ of Hurwitz zeta function at negative integers, which we prove are exponential periods over $\mathbb{E} \left ( w,e^{-wt}\right )$. They can be expressed using Bendersky Gamma functions $\hat Γ_n$ and the values $\log \hat Γ_n(w)$ for $\Re w >0$ are exponential periods over $\mathbb{E} \left ( w,e^{-wt}\right )$. The logarithms of the determinants of the Laplacian on spheres and lens spaces are exponential periods over $\mathbb{E}$. The values $ζ(2n+1)/π^{2n}$ are exponential periods over $\mathbb{E}$. For a periodic function $χ:\mathbb{Z}\to \mathbb{C}$, we extend Ramanujan summation to the twisted series $\sum_{k\geq 1} χ(k) P(k)\log Q(k)$ and define General Twisted Stirling-Ramanujan constants $S_χ(P,Q)$. We derive integral formulas proving that they are exponential periods over $\mathbb{E}(χ)=\mathbb{Q}(χ)(t,e^{-t})$. For $n\geq 0$, $L_χ'(-n)$ and $L_χ(n+1)/π^n$ are given in terms of these constants and are exponential periods over $\mathbb{E}(χ)$.
Comments47 pages. A couple of results added
