Related papers: Dirichlet product of derivative arithmetic with an…
This is a survey of a connection between the distribution of certain power residues modulo $p$, $p$ a prime, and relative class numbers. The focus lies on quadratic residues and sixth power residues. Dirichlet's class number formula yields…
I present a bijection on integer partitions that leads to recursive expressions, closed formulae and generating functions for the cardinality of certain sets of partitions of a positive integer $n$. The bijection leads also to a product on…
Two sums over the inverse of the product of an integer n and its greatest prime factor G(n), are computed to first 13 decimal digits. These sums converge, but converge very slowly. They are transformed into sums involving Mertens' prime…
We define a function by refining Stern's diatomic sequence. We name it the {\it assembly function}. It is strictly increasing continuous. The first and the second main theorems are on an action to the function. The third theorem is on…
We prove that certain quotients of entire functions are characteristic functions. Under some conditions, the probability measure corresponding to a characteristic function of that type has a density which can be expressed as a generalized…
In the paper we describe derivations of some classes of Leibniz algebras. It is shown that any derivation of a simple Leibniz algebra can be written as a combination of three derivations. Two of these ingredients are a Lie algebra…
In this article we study in depth the Dirichlet theorem, which states that if a, b are relative prime integers, the sequence p = an + b contains infinite prime numbers, we simplify and generalize this theorem, we enunciate some special…
On the set of mappings of the given set, we define the product of mappings. If A is associative algebra, then we consider the set of matrices, whose elements are linear mappings of algebra A. In algebra of matrices of linear mappings we…
In this work at first the relation the Mittag-Lefler function to the exponential is given. The results are applied to the construction of the solution of Cauchy problem for ordinary linear operator differential equations with constant…
We set the main concepts for multiplicative fractional calculus. We define Caputo, Riemann and Letnikov multiplicative fractional derivatives and multiplicative fractional integrals and study some of their properties. Finally, the…
We conservatively extend classical elementary differential calculus to the Cartesian closed category of convergence spaces. By specializing results about the convergence space representation of directed graphs, we use Cayley graphs to…
The Wiener index is a graphical invariant that has found extensive application in chemistry. We define a generating function, which we call the Wiener polynomial, whose derivative is a q-analog of the Wiener index. We study some of the…
We investigate the approximation of quadratic Dirichlet $L$-functions over function fields by truncations of their Euler products. We first establish representations for such $L$-functions as products over prime polynomials times products…
In this paper we will discuss the Dirichlet problem of nonlinear second order partial differential equations resolved with any derivatives. First, we transform it into generalized integral equations. Next, we discuss the existence of the…
The primary objective of this paper is to employ methods from analytic number theory to investigate the mean value properties of a composite function involving the Dirichlet divisor function and a generalized minimal power function.…
We study the problem of the product property for the Lempert function with many poles and consider some properties of this function mostly for plane domains.
Extending a classical estimate of Mertens for the sum of the reciprocals of the first primes, we provide an explicit remainder formula for products of an arbitrary, but fixed, number of primes.
A Dirichlet-type problem is studied for an equation of even order with variable coefficients. A criterion for the uniqueness of a solution is given. The solution is built in the form of a Fourier series. When justifying the convergence of…
It is be shown that the sequence of Bernstein polynomials for a function of several variables converges to this function uniformly along with every partial derivative of any order, provided that the latter derivative is well defined and…
This paper introduces DD calculus and describes the basic calculus concepts of derivative and integral in a direct and non-traditional way, without limit definition: Derivative is computed from the point-slope equation of a tangent line and…