[Paper Review] On the behavior of $p$-adic Euler $\ell$-functions

Min-Soo Kim|arXiv (Cornell University)|Oct 11, 2010

Advanced Mathematical Identities20 references3 citations

TL;DR

This paper constructs a $p$-adic Euler $\ell$-function using Kubota-Leopoldt's $p$-adic interpolation and Washington's $p$-adic integral approach, proving its existence as a continuous $p$-adic function on $\mathbb{Z}_p$. The key result establishes that $\ell_{p,E}(1-n,\chi) = (1 - p^n\chi_n(p))\ell_E(-n,\chi_n)$, linking the $p$-adic $\ell$-function to generalized Euler numbers and enabling $p$-adic interpolation of special values.

ABSTRACT

In this paper we propose a construction of $p$-adic Euler $\ell$-function using Kubota-Leopoldt's approach and Washington's one. We also compute the derivative of $p$-adic Euler $\ell$-function at $s=0$ and the values of $p$-adic Euler $\ell$-function at positives integers.

Motivation & Objective

• To construct a $p$-adic Euler $\ell$-function that interpolates special values of the complex Euler $\ell$-function at negative integers.
• To extend the classical Euler $\ell$-function to the $p$-adic setting using $p$-adic integration and interpolation techniques.
• To compute the derivative of the $p$-adic Euler $\ell$-function at $s=0$ and its values at positive integers.
• To establish the $p$-adic continuity and interpolation properties of the $\ell_{p,E}(s,\chi)$ function using $p$-adic measures and character sums.

Proposed method

• Uses the fermionic $p$-adic measure $\mu_{-1}$ on $\mathbb{Z}_p^\times$ to define $p$-adic integrals of Dirichlet characters.
• Applies Washington’s $p$-adic integral representation to express generalized Euler numbers $E_{n,\chi}$ as $\int_{X^*} \chi(x) x^n d\mu_{-1}(x)$.
• Defines the $p$-adic Euler $\ell$-function as $\ell_{p,E}(s,\chi) = \int_{X^*} \chi(x) \langle x \rangle^{1-s} d\mu_{-1}(x)$ for $s \in \mathbb{Z}_p$.
• Utilizes the identity $\ell_{p,E}(1-n,\chi) = (1 - p^n \chi_n(p)) \ell_E(-n, \chi_n)$ to link $p$-adic and complex $\ell$-functions.
• Employs the Kummer congruences and $p$-adic continuity to prove the existence and uniqueness of the $p$-adic $\ell$-function.
• Applies the $p$-adic Mellin transform interpretation to show analytic continuation and $p$-adic interpolation of special values.

Experimental results

Research questions

• RQ1 Can a $p$-adic Euler $\ell$-function be constructed that interpolates $\ell_E(-n, \chi_n)$ for $n \geq 0$?
• RQ2 How can the $p$-adic $\ell$-function be defined using $p$-adic integrals and measures?
• RQ3 What is the value of the derivative of the $p$-adic $\ell$-function at $s = 0$?
• RQ4 What are the values of the $p$-adic $\ell$-function at positive integers?
• RQ5 How do the $p$-adic and complex $\ell$-functions relate via the interpolation formula?

Key findings

• The $p$-adic Euler $\ell$-function $\ell_{p,E}(s,\chi)$ exists and is uniquely defined as a continuous function on $\mathbb{Z}_p$.
• The function satisfies $\ell_{p,E}(1-n,\chi) = (1 - p^n \chi_n(p)) \ell_E(-n, \chi_n)$ for all $n \geq 1$, establishing $p$-adic interpolation.
• The derivative of $\ell_{p,E}(s,\chi)$ at $s = 0$ is computed explicitly via $p$-adic integral techniques.
• The values of $\ell_{p,E}(s,\chi)$ at positive integers are derived using the $p$-adic integral representation and character sums.
• The $p$-adic absolute value of Euler numbers satisfies $|E_n|_p \leq 1$, ensuring integrality in $\mathbb{Z}_p$.
• The construction confirms the Kummer congruences for generalized Euler numbers via $p$-adic continuity of the integral.