On finite analogues of Euler's constant
Masanobu Kaneko, Toshiki Matsusaka, Shin-ichiro Seki
TL;DR
This paper defines finite analogues of Euler's constant within the $\mathcal{A}$-model of finite multiple zeta values, notably $\gamma_\mathcal{A}^\mathrm{W}$ from Wilson quotients and $\gamma_\mathcal{A}^\mathrm{M}$ from Mascheroni/Gregory-type series, and introduces a related family $\gamma_\mathcal{A}^{\mathrm{K},m}$. It proves that the differences among these analogues lie in the $\mathbb{Q}$-span of $1$ and $\log_\mathcal{A}(j)$, with a key identity $\gamma_\mathcal{A}^\mathrm{M} = \gamma_\mathcal{A}^\mathrm{W} + \ell_\mathcal{A}(2) - 1$ linking the two main constructions and connecting Gregory coefficients $G_n$ to $\ell_\mathcal{A}$. An auxiliary interlude expresses $G_\mathcal{A}(k)$ in terms of $\ell_\mathcal{A}$, and the results generalize to Kluyver-type sums, showing the finite analogues are not independent constants but are interwoven with a logarithmic structure in $\mathcal{A}$. The discussion touches on periods in $\mathcal{A}$, Rosen’s perspective on such periods, and potential implications under the ABC-conjecture, highlighting the arithmetic richness of finite analogues of classical constants.
Abstract
We introduce and study finite analogues of Euler's constant in the same setting as finite multiple zeta values. We define a couple of candidate values from the perspectives of a ``regularized value of $ζ(1)$'' and of Mascheroni's and Kluyver's series expressions of Euler's constant using Gregory coefficients. Moreover, we reveal that the differences between them always lie in the $\mathbb{Q}$-vector space spanned by 1 and values of a finite analogue of logarithm at positive integers.