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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⩾1τ(n)e2πinx/n$\sum _{n\geqslant 1} \tau (n) {\rm e}^{2\pi i n x}/\sqrt {n}$ (x∈Q$x\in {\mathbb {Q}}$), 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)$SL(2,{\mathbb {Z}})$ have a Gaussian distribution, and give the first proof of an estimate for 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. [ABSTRACT FROM AUTHOR]

We investigate the first moment of primes in progressions ... as x,N → ∞. We show unconditionally that, when a = 1, there is a significant bias towards negative values, uniformly for N ≼ ec√log x. The proof combines recent results of the authors on the first moment and on the error term in the dispersion method. More generally, for a ∈ ℤ \ {0} we prove estimates that take into account the potential existence (or inexistence) of Landau--Siegel zeros. [ABSTRACT FROM AUTHOR]

We prove that the existence of exceptional real zeros of Dirichlet L-functions would lead to cancellations in the sum Σp≤x Kl(1, p) of Kloosterman sums over primes, and also to sign changes of Kl(1, n), where n runs over integers with exactly two prime factors. Our arguments involve a variant of Bombieri's sieve, bounds for twisted sums of Kloosterman sums, and work of Fouvry and Michel on sums of |Kl(1, n)|. [ABSTRACT FROM AUTHOR]

We show that smooth-supported multiplicative functions f are well-distributed in arithmetic progressions a1a2-1(modq) on average over moduli q≤x3/5-ε with (q,a1a2)=1. [ABSTRACT FROM AUTHOR]

We prove a bound for quintilinear sums of Kloosterman sums, with congruence conditions on the 'smooth' summation variables. This generalizes classical work of Deshouillers and Iwaniec, and is key to obtaining power-saving error terms in applications, notably the dispersion method. As a consequence, assuming the Riemann hypothesis for Dirichlet [ABSTRACT FROM AUTHOR]

A number is said to be -friable if it has no prime factor greater than . In this paper, we prove a central limit theorem on average for the distribution of divisors of -friable numbers less than , for all satisfying . This was previously known under the additional constraint , by work of Basquin. Our argument relies on the two-variable saddle-point method. [ABSTRACT FROM AUTHOR]

We study the mean value of generalized divisor functions τ κ(n) over integers without large prime factors (here κ > 0). We relate this problem to the computation of the ratio Ψ(x1/κ,y) κ/Ψ(x,y), involving the y-smooth numbers counting function. We establish an inverse theorem, giving limitations on the range in (x,y) in which this ratio can be asymptotically estimated uniformly in κ. [ABSTRACT FROM AUTHOR]

An integer $n$ is said to be $y$-friable if its largest prime factor $P^{+}(n)$ is less than $y$. In this paper, it is shown that the $y$-friable integers less than $x$ have a weak exponent of distribution at least $3/5-{\it\varepsilon}$ when $(\log x)^{c}\leqslant x\leqslant x^{1/c}$ for some $c=c({\it\varepsilon})\geqslant 1$, that is to say, they are well distributed in the residue classes of a fixed integer $a$, on average over moduli ${\leqslant}x^{3/5-{\it\varepsilon}}$ for each fixed $a\neq 0$ and ${\it\varepsilon}>0$. We apply this to the estimation of the sum $\sum _{2\leqslant n\leqslant x,P^{+}(n)\leqslant y}{\it\tau}(n-1)$ when $(\log x)^{c}\leqslant y$. This follows and improves on previous work of Fouvry and Tenenbaum. Our proof combines the dispersion method of Linnik in the setting of Bombieri, Fouvry, Friedlander and Iwaniec with recent work of Harper on friable integers in arithmetic progressions. [ABSTRACT FROM AUTHOR]

In a recent paper [5], Lagarias and Soundararajan study the y-smooth solutions to the equation a+b=c. Conditionally under the Generalised Riemann Hypothesis, they obtain an estimate for the number of those solutions weighted by a compactly supported smooth function, as well as a lower bound for the number of bounded unweighted solutions. In this paper, we prove a more precise conditional estimate for the number of weighted solutions that is valid when y is relatively large with respect to x, so as to connect our estimate with the one obtained by La Bretèche and Granville in a recent work [2]. We also prove, conditionally under the Generalised Riemann Hypothesis, the conjectured upper bound for the number of bounded unweighted solutions, thus obtaining its exact asymptotic behaviour. [ABSTRACT FROM AUTHOR]