The p-adic zeta function and a p-adic Euler constant

Abstract

We study the p-adic analogue $\gamma_p$ of the Euler-Mascheroni constant $\gamma$, also known as Euler constant. The p-adic Euler constant can be defined using the p-adic analogue of the gamma function. The constant γp can also be expressed in terms of the Kubota-Leopoldt p-adic L-function: $\gamma_p$ is the constant term in the Laurent series expansion of the $\chi = 1$ branch of the p-adic zeta function about $s = 1$. The p-adic zeta function can be constructed using p-adic distributions or measures and there are different series expansions. We derive several formulas for $\gamma_p$ and present computations with SageMath