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/F of number fields, a Gross-Stark unit is defined as a p-unit of H satisfying certain properties. Let tau is an element of Gal(H/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 Y-p(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) and prove that two refinements are consistent.