A p-adic L-function is a -adic analytic function or measure interpolating suitably adjusted special values of an L-function. The Kubota-Leopoldt p-adic L-function is the basic example attached to a Dirichlet character.
For an even Dirichlet character , the Kubota-Leopoldt function interpolates generalized Bernoulli values with the relevant Euler factor and Teichmüller character adjustment. If , it can be encoded by an integral power series. For , the -ramified convention is . The trivial character requires separate treatment of the pole.
On , this element of the total quotient ring of the Iwasawa algebra becomes an integral measure after multiplication by every difference . Its even moments interpolate the displayed Euler-corrected zeta values. Its trivial-character component has a pole, so the pseudomeasure itself must not be treated as an element of the integral group algebra. Multiplication by the augmentation ideal removes the pole.
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P-adic L-functions are a concept in number theory and algebraic geometry that arises in the study of p-adic numbers and L-functions. They are closely related to both classical L-functions (like Riemann zeta functions and Dirichlet L-functions) and p-adic analysis.