On a $p$-adic analogue of Shintani’s formula

Abstract

Shintani expressed the first derivative at $s = 0$ of a partial $\zeta$-function of an algebraic number field in terms of the multiple gamma function. Cassou-Noguès constructed a $p$-adic analogue of the partial $\zeta$-function and calculated the derivative at $s = 0$. In this paper, we will define a $p$-adic analogue of the multiple gamma function and give a $p$-adic analogue of Shintani’s formula. This formula has a strong resemblance to the original Shintani’s formula. Using this formula, we get a partial result toward Gross’ conjecture concerning the order at $s = 0$ of the $p$-adic $L$-function.