# Linear correlations of the divisor function

## TL;DR

This paper studies the analytic continuation of a divisor function series constrained by a linear equation, revealing its meromorphic structure and applying it to count rational points on a specific algebraic variety.

## Contribution

It extends the understanding of divisor function series by providing their meromorphic continuation and applies this to count solutions on a bihomogeneous variety.

## Key findings

- Series converges for Re(s)>1-1/k
- Meromorphic continuation to Re(s)>1-2/(k+1) with specific poles
- Asymptotic formula with power saving error for rational points on the variety

## Abstract

Motivated by arithmetic applications on the number of points in a bihomogeneous variety and on moments of Dirichlet $L$-functions, we provide analytic continuation for the series $\mathcal A_{\boldsymbol{a}}(s):=\sum_{n_1,\dots,n_k\geq1}\frac{d(n_1)\cdots d(n_k)}{(n_1\cdots n_k)^{s}}$ with the sum restricted to solutions of a non-trivial linear equation $a_1n_1+\cdots+a_kn_k=0$. The series $\mathcal A_{\boldsymbol{a}}(s)$ converges absolutely for $\Re(s)>1-\frac1k$ and we show it can be meromorphically continued to $\Re(s)>1-\frac 2{k+1}$ with poles at $s=1-\frac1{k-j}$ only, for $1\leq j< (k-1)/2$. As an application, we obtain an asymptotic formula with power saving error term for the number of points in the variety $a_1x_1y_1+\cdots+a_kx_ky_k=0$ in $\mathbb P^{k-1}(\mathbb Q)\times \mathbb P^{k-1}(\mathbb Q)$.