General Stirling-Ramanujan Constants are exponential periods and applications

Abstract

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 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 ${\sum }_{k\ge 1}P(k)\log Q(k)$. They are exponential periods over ${\mathbb{K}}_P((\omega ))(t,e^{-t},(e^{-\omega t}))$ 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 ${\zeta }_H^{\prime }(-n,w)$ 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 ${\hat{\Gamma }}_n$ and the values $\log {\hat{\Gamma }}_n(w)$ 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 $\zeta (2n+1)/{\pi }^{2n}$ are exponential periods over $\mathbb{E}$. 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 $\mathbb{E}(\chi )=\mathbb{Q}(\chi )(t,e^{-t})$. For $n\ge 0$, $L_{\chi }^{\prime }(-n)$ and $L_{\chi }(n+1)/{\pi }^n$ are given in terms of these constants and are exponential periods over $\mathbb{E}(\chi )$.

47 pages. A couple of results added