General Stirling-Ramanujan Constants are exponential periods and applications
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 $\mathbb{E}(χ)$.