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2005 On a $p$-adic analogue of Shintani’s formula
Tomokazu Kashio
J. Math. Kyoto Univ. 45(1): 99-128 (2005). DOI: 10.1215/kjm/1250282969

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.

Citation

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Tomokazu Kashio. "On a $p$-adic analogue of Shintani’s formula." J. Math. Kyoto Univ. 45 (1) 99 - 128, 2005. https://doi.org/10.1215/kjm/1250282969

Information

Published: 2005
First available in Project Euclid: 14 August 2009

zbMATH: 1088.11086
MathSciNet: MR2138802
Digital Object Identifier: 10.1215/kjm/1250282969

Subjects:
Primary: 11R42
Secondary: 11M41, 11S40

Rights: Copyright © 2005 Kyoto University

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Vol.45 • No. 1 • 2005
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