Arithmetic Progressions of Squares and Multiple Dirichlet Series

TL;DR

This paper investigates a two-variable Dirichlet series counting primitive three-term arithmetic progressions of squares, establishing its meromorphic continuation and deriving asymptotic counts for such progressions and rational points on a specific quadratic curve.

Contribution

It introduces a novel multiple Dirichlet series for counting arithmetic progressions of squares and proves its meromorphic continuation, enabling new counting results.

Findings

Meromorphic continuation of the Dirichlet series to a02^2.

Asymptotic counts for arithmetic progressions of squares.

Results on rational points on the curve x^2 + y^2 = 2.

Abstract

We study a Dirichlet series in two variables which counts primitive three-term arithmetic progressions of squares. We show that this multiple Dirichlet series has meromorphic continuation to $\mathbb{C}^2$ and use Tauberian methods to obtain counts for arithmetic progressions of squares and rational points on $x^2+y^2=2$.