Large sieve inequalities with power moduli and Waring's problem

We improve the large sieve inequality with $k$th-power moduli, for all $k\ge 5$. Our method relates these inequalities to a restricted variant of Waring's problem. Firstly, we input a classical divisor bound on the number of representations of a positive integer as a sum of two $k$th-powers. Secondly, we input a recent and general result of Wooley on mean values of exponential sums. Lastly, we state a conditional result, based on the conjectural Hardy-Littlewood formula for the number of representations of a large positive integer as a sum of $k+1$ $k$th-powers.Comment: 10 page

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

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 Vall\'ee 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 show that central values of the Esterman function ($L$ function of the divisor function twisted by an additive character) tend to have a Gaussian distribution, with a large variance. We give a dynamical, "trace formula free" proof that central modular symbols associated with a holomorphic cusp form for $SL(2,{\bf Z})$ have a Gaussian distribution. We also recover a result of Vardi on the convergence of Dedekind sums to a Cauchy law, using dynamical methods

Construction of the beta distributions using the random permutation divisors

A subset of cycles comprising a permutation σ in the symmetric group Sn, n ∈ N, is called a divisor of σ. Then the partial sums over divisors with sizes up to un, 0 ≤ u ≤ 1, of values of a nonnegative multiplicative function on a random permutation define a stochastic process with nondecreasing trajectories. When normalized the latter is just a random distribution function supported by the unit interval. We establish that its expectations under various weighted probability measures defined on the subsets of Sn are quasihypergeometric distribution functions. Their limits as n -> 1 cover the class of two-parameter beta distributions. It is shown that, under appropriate conditions, the convergence rate is of the negative power of n order

Generalised divisor sums of binary forms over number fields

Estimating averages of Dirichlet convolutions $1 \ast \chi$, for some real Dirichlet character $\chi$ of fixed modulus, over the sparse set of values of binary forms defined over $\mathbb{Z}$ has been the focus of extensive investigations in recent years, with spectacular applications to Manin's conjecture for Ch\^atelet surfaces. We introduce a far-reaching generalization of this problem, in particular replacing $\chi$ by Jacobi symbols with both arguments having varying size, possibly tending to infinity. The main results of this paper provide asymptotic estimates and lower bounds of the expected order of magnitude for the corresponding averages. All of this is performed over arbitrary number fields by adapting a technique of Daniel specific to $1\ast 1$. This is the first time that divisor sums over values of binary forms are asymptotically evaluated over any number field other than $\mathbb{Q}$. Our work is a key step in the proof, given in subsequent work, of the lower bound predicted by Manin's conjecture for all del Pezzo surfaces over all number fields, under mild assumptions on the Picard number