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arXiv · 2201.08870

Sum Expressions for Kubota-Leopoldt $p$-adic $L$-functions

Luochen Zhao
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Abstract

When $p$ is an odd prime, Delbourgo observed that any Kubota-Leopoldt $p$-adic $L$-function, when multiplied by an auxiliary Euler factor, can be written as an infinite sum. We shall establish such expressions without restriction on $p$, and without the Euler factor when the character is nontrivial, by computing the periods of appropriate measures. As an application, we will reprove the Ferrero-Greenberg formula for the derivative $L_p'(0,\chi)$. We will also discuss the convergence of sum expressions in terms of elementary $p$-adic analysis, as well as their relation to Stickelberger elements; such discussions in turn give alternative proofs of the validity of sum expressions.

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Luochen Zhao. 2022-01-21. Sum Expressions for Kubota-Leopoldt $p$-adic $L$-functions. https://arxiv.org/abs/2201.08870

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