The cubic Pell equation $L$-function
Volume 210 / 2023
Abstract
For $d \gt 1$ a cubefree rational integer, we define an $L$-function (denoted $L_d(s)$) whose coefficients are derived from the cubic theta function for $\mathbb Q(\sqrt {-3})$. The Dirichlet series defining $L_d(s)$ converges for ${\rm Re}(s) \gt 1$, and its coefficients vanish except at values corresponding to integral solutions of $mx^3 - dny^3 = 1$ in $\mathbb Q(\sqrt {-3})$, 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 ${\rm Re}(s) \gt \frac {1}{2}$ and prove that away from its poles, it satisfies the bound $L_d(s) \ll |s|^{{7}/{2}}$ 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 ${\rm Re}(s) \gt \frac {1}{2}$ besides the possible simple pole at $s = \frac {2}{3}$.
