Related papers: On the error term of a lattice counting problem, I…
We improve the error terms of some estimates related to counting lattices from recent work of L. Fukshansky, P. Guerzhoy and F. Luca (2017). This improvement is based on some analytic techniques, in particular on bounds of exponential sums…
In various papers the authors have derived asymptotics for moments of certain cotangent sums related to the Riemann Hypothesis. S. Bettin has given an upper bound for the error term in these asymptotic results. In the present paper the…
We obtain asymptotic formulae with optimal error terms for the number of lattice points under and near a dilation of the standard parabola, the former improving upon an old result of Popov. These results can be regarded as achieving the…
In this note, an upper bound for the sum of fractional parts of certain smooth functions is established. Such sums arise naturally in numerous problems of analytic number theory. The main feature is here an improvement of the main term due…
We give an improvement of sharp Berezin type bounds on the Riesz means $\sum_k(\Lambda-\lambda_k)_+^\sigma$ of the eigenvalues $\lambda_k$ of the Dirichlet Laplacian in a domain if $\sigma\geq 3/2$. It contains a correction term of the…
We examine the size of $E_{2}(T)$, the error term in the asymptotic formula for $\int_{0}^{T} |\zeta(1/2 + it)|^{4}\, dt$ where $\zeta(s)$ is the Riemann zeta-function. We make improvements in the powers of $\log T$ in the known bounds for…
We use classical methods from analytic number theory to resolve the lattice point counting problem on the first Heisenberg group, in the case where the gauge function is taken to be the Cygan-Kor$\acute{a}$nyi Heisenberg-norm…
In this paper, we obtain sharp estimates for the number of lattice points under and near the dilation of a general parabola, the former generalizing an old result of Popov. We apply Vaaler's lemma and the Erd\H{o}s-Turan inequality to…
Assuming the Riemann Hypothesis, we derive explicit bounds for the error terms in short interval analogues of the prime number theorem and Mertens' theorems using a smoothing argument. Our results improve upon previous bounds in both…
Assuming the Generalized Riemann Hypothesis, we obtain a lower bound within a constant factor of the conjectured asymptotic result for the second moment for primes in an individual arithmetic progression in short intervals. Previous results…
Weyls law is a fundamental result governing the asymptotic behaviour of the eigenvalues of teh Laplacian. It states that for a compact d dimensional manifold M (without boundary), the eigenvalue counting function has an asymptotic growth,…
In this paper, we present explicit and computable error bounds for the asymptotic expansions of the Hermite polynomials with Plancherel--Rotach scale. Three cases, depending on whether the scaled variable lies in the outer or oscillatory…
Iosevich and Wyman have proved in ~\cite{IoWy} that the remainder term in classical Weyl law can be improved from $O(\lambda^{d-1})$ to $o(\lambda^{d-1})$ in the case of product manifold by using a famous result of Duistermaat and…
We establish an error estimate for counting lattice points in Euclidean norm balls (associated to an arbitrary irreducible linear representation) for lattices in simple Lie groups of real rank at least two. Our approach utilizes refined…
We estimate the error term in the asymptotic formula of the Sidon constant for (ordinary) Dirichlet polynomials by providing explicit lower and upper bounds. The lower bound is already implicitly known, but we supply the necessary…
In this paper we improve the estimate for the remainder term in the asymptotic formula concerning the circle problem in an arithmetic progression.
We remove logarithmic factors in error term estimates in asymptotic formulas for the number of solutions of a class of additive congruences modulo a prime number.
Asymptotic expansions are derived for solutions of the parabolic cylinder and Weber differential equations. In addition the inhomogeneous versions of the equations are considered, for the case of polynomial forcing terms. The expansions…
We prove uniform versions of two classical results in analytic number theory. The first is an asymptotic for the number of points of a complete lattice $\Lambda \subseteq \mathbb{R}^d$ inside the $d$-sphere of radius $R$. In contrast to…
In this paper, we improve the error term in a previous paper on an asymptotic formula for the number of squarefull numbers in an arithmetic progression.