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Sum expressions for Kubota–Leopoldt $p$
-adic $L$
-functions
- Part of
-
- Journal:
- Proceedings of the Edinburgh Mathematical Society / Volume 65 / Issue 2 / May 2022
- Published online by Cambridge University Press:
- 10 June 2022, pp. 460-479
-
- Article
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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 non-trivial, 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.
