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

The aim of this work is to illustrate a conditional result involving the exponential sums over primes in short intervals under the assumption that both the Generalized Riemann Hypothesis and the Density Hypothesis for Dirichlet…

In this paper we produce precise large deviation estimates through the lens of mod-Poisson convergence. We apply a general result to various examples from number theory, Dedekind domains and polynomials over finite fields when an element is…

Let $\Omega$ be an open, simply connected, and bounded region in $\mathbb{R}^{d}$, $d\geq2$, and assume its boundary $\partial\Omega$ is smooth. Consider solving the eigenvalue problem $Lu=\lambda u$ for an elliptic partial differential…

An important line of research is the investigation of the laws of random variables known as Dirichlet means as discussed in Cifarelli and Regazzini(1990). However there is not much information on inter-relationships between different…

This paper develops an analytic theory of Dirichlet series in several complex variables which possess sufficiently many functional equations. In the first two sections it is shown how straightforward conjectures about the meromorphic…

A new method for numerical solving of boundary problem for ordinary differential equations with slowly varying coefficients which is aimed at better representation of solutions in the regions of their rapid oscillations or exponential…

This paper evaluates some generalised Euler sums involving the digamma function.

We study the sup-norm and mean-square-norm problems for Eisenstein series on certain arithmetic hyperbolic orbifolds, producing sharp exponents for the modular surface and Picard 3-fold. The methods involve bounds for Epstein zeta…

Solutions of the Dirichlet and Robin boundary value problems for the multi-term variable-distributed order diffusion equation are studied. A priori estimates for the corresponding differential and difference problems are obtained by using…

We define the generalized Dirichlet beta and Riemann zeta functions in terms of the integrals, involving powers of the hyperbolic secant and cosecant functions. The corresponding functional equations are established. Some consequences of…

In this paper we consider the distribution of large central values of Dirichlet L-functions over cosets of the group of characters modulo q via Soundararajan's resonator method.

We consider several old problems involving the number of prime divisors function $\omega(n)$, as well as the related functions $\Omega(n)$ and $\tau(n)$. Firstly, we show that there are infinitely many positive integers $n$ such that…

Resummation methods using continued functions are implemented to converge divergent series appearing in perturbation problems related to continuous phase transitions in field theories. In some cases, better convergence properties are…

In this paper we obtain a mean value theorem for a general Dirichlet series $f(s)= \sum_{j=1}^\infty a_j n_j^{-s}$ with positive coefficients for which the counting function $A(x) = \sum_{n_{j}\le x}a_{j}$ satisfies $A(x)=\rho x +…

Explicit bounds on the tails of the zeta function $\zeta$ are needed for applications, notably for integrals involving $\zeta$ on vertical lines or other paths going to infinity. Here we bound weighted $L^2$ norms of tails of $\zeta$. Two…

We apply the resonance method to Montgomery's convolution formula for $\textrm{Re}\left(e^{-i\theta}\log\zeta(\sigma+it)\right)$ in the strip $1/2 < \sigma < 1$. This gives new insight into maximal values of…

For shape optimization problems, governed by elliptic equations with Dirichlet boundary condition and random coefficients, we utilize a penalization technique to get the approximate problem. We consider that uncertainties exists in the…

In the present paper, an algorithm for the numerical solution of the external Dirichlet generalized harmonic problem for a sphere by the method of probabilistic solution (MPS) is given, where generalized indicates that a boundary function…

Integration operational matrix methods based on Zernike polynomials are used to determine approximate solutions of a class of non-homogeneous partial differential equations (PDEs) of first and second order. Due to the nature of the Zernike…

We provide new upper bounds for sums of certain arithmetic functions in many variables at polynomial arguments and, exploiting recent progress on the mean-value of the Erd\H os-Hooley $\Delta$-function, we derive lower bounds for the…