Related papers: The asymptotic in Waring's problem over function f…
Using the circle method, we count integer points on complete intersections in biprojective space in boxes of different side length, provided the number of variables is large enough depending on the degree of the defining equations and…
Let $\mathbb{F}_q[t]$ be the ring of polynomials over $\mathbb{F}_q$, the finite field of $q$ elements, and let $p$ be the characteristic of $\mathbb{F}_q$. We denote $\widetilde{G}_q(k)$ to be the least integer $t_0$ with the property that…
We obtain estimates for Vinogradov's integral which for the first time approach those conjectured to be the best possible. Several applications of these new bounds are provided. In particular, the conjectured asymptotic formula in Waring's…
A classical problem in number theory is showing that the mean value of an arithmetic function is asymptotic to its mean value over a short interval or over an arithmetic progression, with the interval as short as possible or the modulus as…
We study the arithmetic analogue of maximal functions on diagonal hypersurfaces. This paper is a natural step following the papers of \cite{Magyar_dyadic}, \cite{Magyar_ergodic} and \cite{MSW}. We combine more precise knowledge of…
This paper studies the minimal conditions under which we can establish asymptotic formulae for Waring's problem and other additive problems that may be tackled by the circle method. We confirm in quantitative terms the well-known heuristic…
In this paper we obtain sharp results for Waring's problem over general finite rings, by using a combination of Artin-Wedderburn theory and Hensel's lemma and building on new proofs of analogous results over finite fields that are achieved…
A celebrated result of Hal\'asz describes the asymptotic behavior of the arithmetic mean of an arbitrary multiplicative function with values on the unit disc. We extend this result to multilinear averages of multiplicative functions…
Let $k$ be a positive integer and let $X_k$ be the cubic hypersurface defined by the equation $x^3-(y_1^2+\cdots+y_{4k}^2)z=0$. In this paper, we give an asymptotic formula for the counting function of semi-integral points on $X_k$. We also…
We introduce a number field analogue of the Mertens conjecture and demonstrate its falsity for all but finitely many number fields of any given degree. We establish the existence of a logarithmic limiting distribution for the analogous…
We obtain an asymptotic formula for the mean value of L-functions associated to cubic characters over F_q[t]. We solve this problem in the non-Kummer setting when q=2 (mod 3) and in the Kummer case when q=1 (mod 3). The proofs rely on…
We provide a general result for the algebraic independence of Mahler functions by a new method based on asymptotic analysis. As a consequence of our method, these results hold not only over $\mathbb{C}(z)$, but also over…
We discuss how one could study asymptotics of cyclotomic quantities via the mean values of certain multiplicative functions and their Dirichlet series using a theorem of Delange. We show how this could provide a new approach to Artin's…
When $A=3$, the positive integral solutions of the so-called Markoff equation $$M_A:x^2 + y^2 + z^2 = Axyz$$ can be generated from the single solution $(1,1,1)$ by the action of certain automorphisms of the hypersurface. Since Markoff's…
We study the number of representations of an integer n=F(x_1,...,x_s) by a homogeneous form in sufficiently many variables. This is a classical problem in number theory to which the circle method has been succesfully applied to give an…
The paper proves a generalization of Wintner's theorem on the asymptotics of summation functions to the case of summation functions with nonlinear asymptotics. The class of arithmetic functions that have a logarithmic asymptotic mean is…
We study the counting function of cubic function fields. Specifically, we derive an asymptotic formula for this counting function including a secondary term and an error term of order $\mathcal{O}\big(X^{2/3+\epsilon}\big)$, which matches…
A classical additive basis question is Waring's problem. It has been extended to integer polynomial and non-integer power sequences. In this paper, we will consider a wider class of functions, namely functions from a Hardy field, and show…
In this paper we introduce a new approach and obtain new results for the problem of studying polynomial images of affine subspaces of finite fields. We improve and generalise several previous known results, and also extend the range of such…
We prove the Effective Bogomolov Conjecture, and so the Bogomolov Conjecture, over a function field of characteristic 0 by proving Zhang's Conjecture about certain invariants of metrized graphs. In the function field case, these conjectures…