General Stirling-Ramanujan Constants are exponential periods and applications

For $n\ge 0$, Stirling-Ramanujan constants $S_n$ are the Ramanujan summation of the divergent series ${\sum }_{k\ge 1}{k}^n\log k$. These constants are exponential periods over the base field $\mathbb{Q}(t,e^{-t})$. We generalize this result to a broad class of General Stirling-Ramanujan constants. Given polynomials $P$ and $Q$, with $\mathrm{\Re }Q(s)>0$ for $\mathrm{\Re }s>0$, the constant $S(P,Q)$ is the Ramanujan summation of the series ${\sum }_{k\ge 1}P(k)\log Q(k)$. They are exponential periods over the base field ${\mathbb{K}}_P((\omega ))(t,e^{-t},(e^{-\omega t}))$ where ${\mathbb{K}}_P$ is the field of definition of $P$ and $(\omega )$ are the zeros of $Q$. These constants are related to the derivatives ${\zeta }_H^{\mathrm{\prime }}(-n,w)$ of Hurwitz zeta function at negative integers, which we prove are exponential periods over the base field $\mathbb{Q}(t,e^{-t},w,e^{-wt})$. They can be expressed using Bendersky Gamma functions ${\widehat{\mathrm{\Gamma }}}_n$ and the values $\log {\widehat{\mathrm{\Gamma }}}_n(w)$ for $\mathrm{\Re }w>0$ are exponential periods over the base field $\mathbb{Q}(t,e^{-t},w,e^{-wt})$. The logarithms of the zeta-regularized determinants of the Laplacian on spheres are exponential periods in the base field $\mathbb{Q}(t,e^{-t})$. The values $\zeta (2n+1)\mathrm{/}{\pi }^{2n}$ are exponential periods in the base field $\mathbb{Q}(t,e^{-t})$. For a periodic function $\chi :\mathbb{Z}\to \mathbb{C}$, we extend Ramanujan summation to the twisted series ${\sum }_{k\ge 1}\chi (k)P(k)\log Q(k)$ and define General Twisted Stirling-Ramanujan constants $S_{\chi }(P,Q)$. We derive integral formulas proving that they are exponential periods over $\mathbb{Q}(\chi )(t,e^{-t})$. For $n\ge 0$, the derivative of the $L$-function, $L_{\chi }^{\mathrm{\prime }}(-n)$ can be expressed in terms of these constants and is an exponential period over the base field $\mathbb{Q}(\chi )(t,e^{-t})$.