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Arithmetic constants for symplectic variances of the divisor function

Vivian Kuperberg, Matilde Lalín

Abstract

In [arXiv:2212.04969], the authors stated some conjectures on the variance of certain sums of the divisor function $d_k(n)$ over number fields, which were inspired by analogous results over function fields proven in [arXiv:2107.01437]. These problems are related to certain symplectic matrix integrals. While the function field results can be directly related to the random matrix integrals, the connection between the random matrix integrals and the number field results is less direct and involves arithmetic factors. The goal of this article is to give heuristic arguments for the formulas of these arithmetic factors.

Arithmetic constants for symplectic variances of the divisor function

Abstract

In [arXiv:2212.04969], the authors stated some conjectures on the variance of certain sums of the divisor function over number fields, which were inspired by analogous results over function fields proven in [arXiv:2107.01437]. These problems are related to certain symplectic matrix integrals. While the function field results can be directly related to the random matrix integrals, the connection between the random matrix integrals and the number field results is less direct and involves arithmetic factors. The goal of this article is to give heuristic arguments for the formulas of these arithmetic factors.

Paper Structure

This paper contains 10 sections, 5 theorems, 113 equations, 4 figures.

Table of Contents

  1. Introduction
  2. The leading term of $\gamma_{d_k,2}^S(c)$
  3. Results on evaluating multiple Dirichlet series
  4. The constant in Conjecture \ref{['conj:symp-square-fund-discs']}
  5. The constant in Conjecture \ref{['conj:symp-square']}
  6. The constant in Conjecture \ref{['conj:symp-rudnickwaxman']}
  7. Discrepancies from the function field case and factors not present in the unitary case
  8. A factor of $2$
  9. The choice of interval
  10. Numerics

Key Result

Proposition 2.1

KuperbergLalin2 For $n\leq N+\frac{1+k}{2}$, we have Moreover, $I_{d_k,2}^S(n;N)$ is a quasi-polynomial in $n$ of period 2 and degree $2k^2+k-2$ (provided that $n\leq N+\frac{1+k}{2}$).

Figures (4)

  • Figure 1: Graph of ratios between the numerical output and our expectation for the variance in Conjecture \ref{['conj:symp-square-fund-discs']} when $\frac{\log x}{\log y} = 0.5$ for the cases $k=1$ (left) and $k=2$ (right).
  • Figure 2: Graph of ratios between the numerical output and our expectation for the variance in Conjecture \ref{['conj:symp-square']} when $\frac{\log x}{\log y} = 0.5$ for the cases $k=1$ (left) and $k=2$ (right).
  • Figure 3: Graph of ratios between the numerical output and our expectation for the variance in Conjecture \ref{['conj:symp-rudnickwaxman']} when $\frac{\log x}{2\log K} = 0.4$ for the cases $\ell=1$ (left) and $\ell=2$ (right).
  • Figure 4: Graph of ratios between the numerical output and our expectation for the variance in Conjecture \ref{['conj:symp-rudnickwaxman']} when $\frac{\log x}{2\log K} = 0.4$ with logarithmic scale for the cases $\ell=1$, basis $1.25$ (left) and $\ell=2$, basis $1.15$ (right).

Theorems & Definitions (9)

  • Conjecture 1.1
  • Conjecture 1.2
  • Conjecture 1.3
  • Proposition 2.1
  • Theorem 3.1
  • Theorem 3.2
  • Remark 3.3
  • Lemma 4.1
  • Lemma 4.2