Counting Divisors in the Outputs of a Binary Quadratic Form

Chan Ieong Kuan; David Lowry-Duda; Alexander Walker; Tinghao Huang

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arXiv · 2310.13632

Counting Divisors in the Outputs of a Binary Quadratic Form

Chan Ieong Kuan·David Lowry-Duda·Alexander Walker·Tinghao Huang

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Abstract

For a fixed natural number $h$, we prove meromorphic continuation of the two-variable Dirichlet series $\sum_m r_2(m) \sigma_w(m + h) (m + h)^{-s + w}$ to $\mathbb{C}^2$ and use this to obtain asymptotics for $\sum_{m^2 + n^2 \leq X} \sigma_w(m^2 + n^2 + h)$. We approach this continuation through spectral theory. Our results are comparable to earlier work of Bykovskii, who used different methods to study the sums $\sum_{n^2 \leq X} \sigma_w(n^2 + h)$.

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Chan Ieong Kuan, David Lowry-Duda, Alexander Walker, Tinghao Huang. 2023-10-20. Counting Divisors in the Outputs of a Binary Quadratic Form. https://arxiv.org/abs/2310.13632

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