On Finite Analogues of Euler’s Constant

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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 $\zeta (1)$” 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.