On finite analogues of Euler's constant

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 $\zeta(1)$'' 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.