$p$-adic measures associated with zeta values and $p$-adic $\log$ multiple gamma functions

We study a relation between two refinements of the rank one abelian Gross-Stark conjecture: For a suitable abelian extension $H\mathrm{/}F$ of number fields, a Gross-Stark unit is defined as a $p$-unit of $H$ satisfying some proporties. Let $\tau \in \mathrm{G}\mathrm{a}\mathrm{l}(H\mathrm{/}F)$. Yoshida and the author constructed the symbol $Y_p(\tau )$ by using $p$-adic $\log$ multiple gamma functions, and conjectured that the ${\log }_p$ of a Gross-Stark unit can be expressed by $Y_p(\tau )$. Dasgupta constructed the symbol $u_T(\tau )$ by using the $p$-adic multiplicative integration, and conjectured that a Gross-Stark unit can be expressed by $u_T(\tau )$. In this paper, we give an explicit relation between $Y_p(\tau )$ and $u_T(\tau )$.