[Paper Review] Variance of sums in arithmetic progressions of arithmetic functions associated with higher degree $L$-functions in $\mathbb{F}_q[t]$

Chris Hall, Jonathan P. Keating+1 more|arXiv (Cornell University)|Mar 27, 2017

Analytic Number Theory Research23 references3 citations

TL;DR

This paper computes the variance of sums of arithmetic functions associated with higher-degree $L$-functions over $$\mathbb{F}_q[t]\u0024$ in arithmetic progressions, as $q \to \infty$, using equidistribution of Frobenius conjugacy classes and evaluating resulting matrix integrals. The key result shows that these variances differ significantly from degree-one $L$-function cases and match expressions predicted in the number field setting under a generalized pair-correlation conjecture.

ABSTRACT

We compute the variances of sums in arithmetic progressions of arithmetic functions associated with certain $L$-functions of degree two and higher in $\mathbb{F}_q[t]$, in the limit as $q o\infty$. This is achieved by establishing appropriate equidistribution results for the associated Frobenius conjugacy classes. The variances are thus related to matrix integrals, which may be evaluated. Our results differ significantly from those that hold in the case of degree-one $L$-functions (i.e. situations considered previously using this approach). They correspond to expressions found recently in the number field setting assuming a generalization of the pair-correlation conjecture. Our calculations apply, for example, to elliptic curves defined over $\mathbb{F}_q[t]$.

Motivation & Objective

• To compute the variance of sums of arithmetic functions associated with higher-degree $L$-functions in arithmetic progressions over $\mathbb{F}_q[t]$.
• To establish equidistribution results for Frobenius conjugacy classes linked to these $L$-functions.
• To evaluate the resulting matrix integrals that govern the variance in the $q \to \infty$ limit.
• To compare the results with known conjectures in the number field setting, particularly generalized pair-correlation predictions.
• To extend the understanding of fluctuations in arithmetic sums beyond the degree-one $L$-function case, relevant to elliptic curves over $\mathbb{F}_q[t]$.

Proposed method

• Use of the Frobenius conjugacy class equidistribution theorems in the function field setting to model statistical behavior of arithmetic functions.
• Application of Tannakian formalism and monodromy group theory to analyze the geometric and arithmetic structures of sheaves associated with $L$-functions.
• Identification of the geometric Frobenius conjugacy classes in the Tannakian group $G_{\omega_{\mathbf{1}}|M}$ for sheaves over $\mathbb{G}_m$.
• Computation of matrix integrals arising from the monodromy groups to determine the asymptotic variance of sums in arithmetic progressions.
• Use of Mellin transforms and middle-extension sheaves to relate $L$-function arithmetic to Galois representations and Frobenius traces.
• Leveraging the $\ell$-adic cohomology framework to analyze the mixed weight structures of sheaf representations and their Frobenius eigenvalues.

Experimental results

Research questions

• RQ1 How do the variances of sums of arithmetic functions in arithmetic progressions behave for higher-degree $L$-functions over $\mathbb{F}_q[t]$ as $q \to \infty$?
• RQ2 What is the role of Frobenius conjugacy class equidistribution in determining these variances?
• RQ3 How do the resulting matrix integrals compare to those in the number field setting under generalized pair-correlation conjectures?
• RQ4 In what ways do the variances for degree-two and higher $L$-functions differ from those in the degree-one case?
• RQ5 To what extent do the results for $\mathbb{F}_q[t]$ mirror or extend known conjectures for the Riemann zeta function and its $L$-function analogues?

Key findings

• The variance of sums in arithmetic progressions for higher-degree $L$-functions in $\mathbb{F}_q[t]$ is asymptotically governed by matrix integrals derived from the monodromy groups of associated sheaves.
• The resulting variance expressions differ significantly from those in the degree-one $L$-function case, which correspond to the classical prime number theorem in short intervals.
• The computed variances match predictions from generalized pair-correlation conjectures in the number field setting, particularly for $\log X \leq h \leq X^{1/2}$.
• The method successfully applies to arithmetic functions associated with elliptic curves over $\mathbb{F}_q[t]$, providing a function field analogue of the Goldston-Montgomery variance conjecture.
• The Frobenius conjugacy classes associated with the $L$-functions are shown to equidistribute in the Tannakian group, enabling the evaluation of statistical moments via invariant theory.
• The results are valid in the limit $q \to \infty$, with the variance asymptotics expressed in terms of $X$, $c$, and arithmetic invariants like $\phi(c)$ and $\sum_{p|c} \frac{\log p}{p-1}$.