The cubic Pell equation L-function

TL;DR

This paper introduces an $L$-function associated with the cubic Pell equation, proves its meromorphic continuation, and analyzes its pole structure, extending methods from trace formulas in number theory.

Contribution

It generalizes the Takhtajan-Vinogradov trace formula to establish the meromorphic continuation of a new $L$-function linked to cubic Pell equations and studies its pole behavior.

Findings

Successfully meromorphically continued $L_d(s)$ to $ ext{Re}(s) > 1/2$

Identified potential poles at $s=2/3$ and zeros of an Appell hypergeometric function

Bounded $L_d(s)$ by $|s|^{7/2}$ away from poles

Abstract

For $d > 1$ a cubefree rational integer, we define an $L$-function (denoted $L_d(s)$) whose coefficients are derived from the cubic theta function for $\mathbb Q\left(\sqrt{-3}\right)$. The Dirichlet series defining $L_d(s)$ converges for $\text{Re}(s) > 1$, and its coefficients vanish except at values corresponding to integral solutions of $mx^3 - dny^3 = 1$ in $\mathbb Q\left(\sqrt{-3}\right)$, where $m$ and $n$ are squarefree. By generalizing the methods used to prove the Takhtajan-Vinogradov trace formula, we obtain the meromorphic continuation of $L_d(s)$ to $\text{Re}(s) > \frac{1}{2}$ and prove that away from its poles, it satisfies the bound $L_d(s) \ll |s|^{\frac{7}{2}}$ and has a possible simple pole at $s = \frac{2}{3}$, possible poles at the zeros of a certain Appell hypergeometric function, with no other poles. We conjecture that the latter case does not occur, so that $L_d(s)$ has no other poles with $\text{Re}(s) > \frac{1}{2}$ besides the possible simple pole at $s = \frac{2}{3}$.