Related papers: Omega Results for The Divisor and Circle Problems …

We establish several new $\Omega$-theorems for logarithmic derivatives of the Riemann zeta function and Dirichlet $L$-functions. In particular, this improves on earlier work of Landau (1911), Bohr-Landau (1913), and recent work of Lamzouri.

We study large values of quadratic character sums with summation lengths exceeding the square root of the modulus. Assuming the Generalized Riemann Hypothesis, we obtain a new Omega result.

Using the Dirichlet integrals, which are employed in the theory of Fourier series, this paper develops a useful method for the summation of series and the evaluation of integrals.

We study and solve the Dirichlet problem for graphs of prescribed mean curvature in $\mathbb R^{n+1}$ over general domains $\Omega$ without requiring a mean convexity assumption. By using pieces of nodoids as barriers we first give…

This is part II of our examination of the second and fourth moments and shifted moments of the Riemann zeta-function on the critical line using long Dirichlet polynomials and divisor correlations.

The research in the subfield of analytic number theory around error term of summation of sigma functions possesses a history which can be dated back to the mid-19th century when Dirichlet provided an $O(\sqrt{n})$ estimation of error term…

We study the Dirichlet boundary value problem for equations with absorption of the form $-\Delta u+g\circ u=\mu$ in a bounded domain $\Omega\subset R^N$ where $g$ is a continuous odd monotone increasing function. Under some additional…

We settle the issue of well-posedness for the Dirichlet problem for a higher order elliptic system ${\mathcal L}(x,D_x)$ with complex-valued, bounded, measurable coefficients in a Lipschitz domain $\Omega$, with boundary data in Besov…

We consider the "Method of particular solutions" for numerically computing eigenvalues and eigenfunctions of the Laplacian $\Delta$ on a smooth, bounded domain Omega in RR^n with either Dirichlet or Neumann boundary conditions. This method…

We consider the semilinear Dirichlet problem \[ \Delta u+kg(u)=\mu _1 \varphi _1+\cdots +\mu _n \varphi _n+e(x) \;\; \mbox{for $x \in \Omega$}, \;\; u=0 \;\; \mbox{on $\partial \Omega$}, \] where $\varphi _k$ is the $k$-th eigenfunction of…

In this paper, the aim of our work is to establish global weighted gradient estimates via fractional maximal functions and the point-wise regularity estimates of Dirichlet problem for divergence elliptic equations of the type \begin{align*}…

This paper is closely related to the recent work [BW17] of the same authors and our purpose is to elaborate more on some of the results and methods from [BW17]. More specifically our goal is two-fold. Firstly, we will indicate how a simple…

We study sums of the shape $\sum_{n \leqslant x} f \left( \lfloor x/n \rfloor \right)$ where $f$ is either the von Mangoldt function or the Dirichlet-Piltz divisor functions. We improve previous estimates when $f = \Lambda$ and $f = \tau$,…

We consider the asymptotic behavior of the mean square of truncations of the Dirichlet series of $\zeta(s)^k$. We discuss the connections of this problem with that of the variance of the divisor function in short intervals and in arithmetic…

For a fixed $\theta\neq 0$, we define the twisted divisor function $$ \tau(n, \theta):=\sum_{d\mid n}d^{i\theta}\ .$$ In this article we consider the error term $\Delta(x)$ in the following asymptotic formula $$ \sum_{n\leq x}^*|\tau(n,…

We study three special Dirichlet series, two of them alternating, related to the Riemann zeta function. These series are shown to have extensions to the entire complex plane and we find their values at the negative integers (or residues at…

We show that a bilinear estimate for biharmonic functions in a Lipschitz domain $\Omega$is equivalent to the solvability of the Dirichlet problem for the biharmonic equationin $\Omega$. As a result, we prove that for any given bounded…

In this paper, we apply the Dirichlet convolution method to \begin{equation*} T_{k}(x)=\sum_{n \leq x} d_{k}(n), \end{equation*} for $k\ge 3$, where $d_{k}(n)$ is the number of ways to represent $n$ as a product of $k$ positive integer…

In this paper we present a method to deal with divergences in perturbation theory using the method of the Zeta regularization, first of all we use the Euler-Mc Laurin Sum formula to associate the divergent integral to a divergent sum in the…

Asymptotic formulae for Titchmarsh-type divisor sums are obtained with strong error terms that are uniform in the shift parameter. This applies to more general arithmetic functions such as sums of two squares, improving the error term in…