Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
134 tokens/sec
GPT-4o
9 tokens/sec
Gemini 2.5 Pro Pro
47 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

On q-series and the moment problem associated to local factors (2403.01247v1)

Published 2 Mar 2024 in math.NT, math.CO, and math.QA

Abstract: We investigate the moment problem and Jacobi matrix associated -- by the operator theoretic framework of the semilocal trace formula -- to each finite set $S$ of places of $\mathbb Q$ containing the archimedean place. The measure is given by the absolute value squared of the product over $S$ of local factors restricted to the critical line. We treat the case $S={p,\infty}$, where a single prime $p$ is adjoined to the archimedean place. We find that all the key ingredients such as the moments, the orthogonal polynomials and the Jacobi matrices can be expressed as power series in terms of the parameter $q:=1/p$. We show that the series which appear for the moments themselves are Lambert series. The study of the $q$-series for the coefficients of the Jacobi matrix, and for the associated orthogonal polynomials reveals an intriguing integrality result: all those coefficients belong to the ring $\mathbb Z[\frac{1}{\sqrt{2}}]$ obtained by adjoining $1 / \sqrt{2}$ to the ring of integers. The main result of this paper is the conceptual explanation of this integrality property using Catalan numbers.

Summary

We haven't generated a summary for this paper yet.