Limit laws for rational continued fractions and value distribution of quantum modular forms

Abstract

We study the limiting distributions of Birkhoff sums of a large class of cost functions (observables) evaluated along orbits, under the Gauss map, of rational numbers in (0, 1] ordered by denominators. We show convergence to a stable law in a general setting, by proving an estimate with power-saving error term for the associated characteristic function. This extends results of Baladi and Vallée on Gaussian behaviour for costs of moderate growth. We apply our result to obtain the limiting distribution of values of several key examples of quantum modular forms. We obtain the Gaussian behaviour of central values of the Esterman function ∑ 𝑛⩾1 𝜏(𝑛)e 2𝜋𝑖𝑛𝑥 ∕ √ 𝑛 (𝑥 ∈ ℚ), a problem for which known approaches based on Eisenstein series have been so far ineffective. We give a new proof, based on dynamical systems, that central modular symbols associated with a holomorphic cusp form for 𝑆𝐿(2, ℤ) have a Gaussian distribution, and give the first proof of an estimatefor their probabilities of large deviations. We also recover a result of Vardi on the convergence of Dedekind sums to a Cauchy law, using dynamical methods.

Wide moments of L-functions I : Twists by class group characters of imaginary quadratic fields

“…By now, there exist precise conjectures for all moments of families of L-functions [Conrey et al 2005] with fascinating connections to random matrix theory [Keating and Snaith 2000]. These moment conjectures are of deep arithmetic importance through their connections to the important topics of nonvanishing and subconvexity (see, e.g., [Blomer et al 2018]), which in turn are connected to, respectively, rational points on elliptic curves (via the B-S-D conjectures, see [Kolyvagin 1988]) and equidistribution problems (via the Waldspurger formula, see [Michel and Venkatesh 2006]).…”

“…The first example in the literature of an asymptotic evaluation of a (higher) wide moment of automorphic L-functions seems to be the work of Bettin [2019] on Dirichlet L-functions (note that here the terminology "iterated moments" is used):…”

“…Using this relation together with orthogonality of characters and equidistribution of Heegner points, Michel and Venkatesh [2007] calculated the first moment of L f ˝ ; 1 2 , which they combined with subconvexity to obtain quantitative nonvanishing for these central values. This idea has since been generalized in many directions to obtain a variety of nonvanishing results [Dittmer et al 2015;Burungale and Hida 2016;Khayutin 2020;Templier 2011a;2011b].…”