Related papers: Harmonic numbers and the prime counting function
Several interesting formulas concerning finite Hilbert transform and logarithmic integrals are proved with application in determining equilibrium measures, planar limits of analytic random matrix models with $1-$cut potential and solving…
Our aim in this article is to obtain the limit of counting function for the Dirichlet eigenvalues involving the m-order logarithmic Laplacian in a bounded Lipschitz domain and to derive also the lower bound.
It is proved that harmonic functions are characterized by harmonicity of their spherical means, for which purpose the iterated spherical means are used. The similar characterization of solutions to the modified Helmholtz equation…
The theory of harmonic based function is discussed here within the framework of umbral operational methods. We derive a number of results based on elementary notions relying on the properties of Gaussian integrals.
Motivared by Carleman's proof of the isoperimetric inequality in the plane, we study some sharp integral inequalities for harmonic functions on the upper halfspace. We also derive the regularity for nonnegative solutions of the associated…
In this paper, we got some refinements of the norm inequalities related to the Heinz mean and logarithmic mean.
We obtain a new decomposition of the Riemann-Liouville operators of fractional integration as a series involving derivatives (of integer order). The new formulas are valid for functions of class $C^n$, $n \in \mathbb{N}$, and allow us to…
Using lattice approximations of Euclidean space, we develop a way to approximate stable processes that are represented by stochastic integrals over Euclidean space. Via a stable version of the Lindeberg-Feller Theorem we show that the…
In this note, we derive a finite summation formula and an infinite summation formula involving Harmonic numbers of order up to some order by means of several definite integrals
This article deals with the numerical approximation of effective coefficients in stochastic homogenization of discrete linear elliptic equations. The originality of this work is the use of a well-known abstract spectral representation…
We present a notion of primitive which corresponds exactly with the Riemann integral. We obtain a characterization of the integrability in the sense of Riemann which produces a Fundamental Theorem of Calculus without special assumptions. We…
Let $X$ be a large parameter. We will first give a new estimate for the integral moments of primes in short intervals of the type $(p,p+h]$, where $p\leq X$ is a prime number and $h=\odi{X}$. Then we will apply this to prove that for every…
We give an exposition of some connections between Fourier optimization problems and problems in number theory. In particular, we present some recent conditional bounds under the generalized Riemann hypothesis, achieved via a Fourier…
We construct a Beurling generalized number system satisfying the Riemann hypothesis and whose integer counting function displays extremal oscillation in the following sense. The prime counting function of this number system satisfies…
The prime-counting function $\pi(x)$ which computes the number of primes smaller or equal to a given real number has a long-standing interest in number theory. The present manuscript proposes a method to compute $\pi(x)$ with time…
We identify a partition-theoretic generalization of Riemann zeta function and the equally positive integer-indexed harmonic sums at infinity, to obtain the generating function and the integral representations of the latter. The special…
In this note we show that conforming Galerkin approximations for p-harmonic functions tend to infinity-harmonic functions in the limit p \to \infty and h \to 0, where h denotes the Galerkin discretisation parameter.
We arrive at some new relations for the prime number $P_n$, based on the logarithmic and absolute-value properties of the function $\pi(x)$.
Jacobsthal's function was recently generalised for the case of paired progressions. It was proven that a specific bound of this function is sufficient for the truth of Goldbach's conjecture and of the prime pairs conjecture as well. We…
We investigate some extremal problems in Fourier analysis and their connection to a problem in prime number theory. In particular, we improve the current bounds for the largest possible gap between consecutive primes assuming the Riemann…