#### Vol. 13, No. 1, 2019

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Variance of arithmetic sums and $L$-functions in $\mathbb{F}_q[t]$

### Chris Hall, Jonathan P. Keating and Edva Roditty-Gershon

Vol. 13 (2019), No. 1, 19–92
DOI: 10.2140/ant.2019.13.19
##### Abstract

We compute the variances of sums in arithmetic progressions of arithmetic functions associated with certain $L$-functions of degree 2 and higher in ${\mathbb{F}}_{q}\left[t\right]$, in the limit as $q\to \infty$. 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-1 $L$-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 ${\mathbb{F}}_{q}\left[t\right]$.

##### Keywords
$L$-functions, Mellin transform
##### Mathematical Subject Classification 2010
Primary: 11T55
Secondary: 11M38, 11M50