On Finite Analogues of Euler's Constant.

The article focuses on defining and analyzing finite analogues of Euler's constant \(\gamma\) within the algebra \(\mathcal{A}\), a quotient ring constructed from the product of residue rings modulo primes. Two primary analogues are introduced: \(\gamma_{\mathcal{A}}^{\mathrm{W}}\), defined via Wilson quotients, and \(\gamma_{\mathcal{A}}^{\mathrm{M}}\), defined using truncated series of Gregory coefficients (Bernoulli numbers of the second kind). The authors establish explicit relations between these analogues and connect them to a finite analogue of the logarithm function \(\log_{\mathcal{A}}\), showing that differences among these analogues lie in the \(\mathbb{Q}\)-vector space spanned by 1 and values of \(\log_{\mathcal{A}}\) at positive integers. Further variations inspired by Kluyver's formulas are also studied, revealing a structured interplay between these finite constants and arithmetic functions in \(\mathcal{A}\). The work situates these finite analogues in the broader context of finite multiple zeta values and conjectural isomorphisms relating \(\mathcal{A}\) to classical multiple zeta value algebras.