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

Abstract

Shintani expressed the first derivative at s = 0 of a partial ζfunction of an algebraic number field in terms of the multiple gamma function. Cassou-Noguès constructed a p-adic analogue of the partial ζ-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.

On p -adic Hurwitz-type Euler zeta functions

“…Next we define the projection x for all x ∈ C × p , as was done by Kashio [9] and by Tangedal and Young in [23].…”

Explicit computation of Gross–Stark units over real quadratic fields

“…By the result of Kashio mentioned in the Introduction [K,Theorem 6.2], the first derivative of the p-adic version ζ mp,p (s, C + ) of ζ mp (s, C + ) evaluated at s = 0 is also given by (12) and (14) combined, once all the terms are interpreted properly in a p-adic manner. By the comment immediately following Eq.…”

“…An important result due to Kashio [K,Theorem 6.2] says in this case that the value ζ S,7 (0, σ 0 ) is given by the same formula as in (4) once the "correct" p-adic interpretation is given to the log gamma function.…”

On p-adic multiple zeta and log gamma functions

“…The case of general ω p 1 may be deduced from the dilation relations (Theorem 3.2(ii) above and [K,Eq. (5.7)]) of ζ p,N and ζ * p,N .…”

“…2 Remark. Since LΓ p,1 (x; 1) = log p Γ p (x) for x ∈ Z p [K,Lemma 5.5], when N = 1 and ω 1 = 1 the second statement in Theorem 3.5 becomes the well-known relation [L, p. 395] …”