Related papers: Arithmetical Functions : Infinite Products
In this note, we take another look at some arithmetic identities of N.J. Fine associated with divisor functions. We connect these functions with indefinite quadratic forms using a result due to Andrews. As a consequence, arithmetic theorems…
In our recent publication we obtained a series expansion of the arctangent function involving complex numbers. In this work we show that this formula can also be expressed as a real rational function.
We study the analog of power series expansions on the Sierpinski gasket, for analysis based on the Kigami Laplacian. The analog of polynomials are multiharmonic functions, which have previously been studied in connection with Taylor…
For any two arithmetic functions $f,g$ let $\bullet$ be the commutative and associative arithmetic convolution $(f\bullet g)(k):=\sum_{m=0}^k \left( \begin{array}{c} k m \end{array} \right)f(m)g(k-m)$ and for any $n\in\mathbb{N},$…
The work is devoted to the construction of a new interval arithmetic which would combine algorithmic efficiency and high quality estimation of the ranges of expressions.
We survey general properties of multiplicative arithmetic functions of several variables and related convolutions, including the Dirichlet convolution and the unitary convolution. We introduce and investigate a new convolution, called gcd…
Function approximation is a generic process in a variety of computational problems, from data interpolation to the solution of differential equations and inverse problems. In this work, a unified approach for such techniques is…
A construction of product measures is given for an arbitrary sequence of measure spaces via outer measure techniques without imposing any condition on the underlying measure spaces. This approach concludes finally the problem of the…
In this article, we consider the weighted partition function $p_f(n)$ given by the generating series $\sum_{n=1}^{\infty} p_f(n)z^n = \prod_{n\in\mathbb{N}^{*}}(1-z^n)^{-f(n)}$, where we restrict the class of weight functions to strongly…
In this paper, we define finitely additive, probability and modular functions over semiring-like structures. We investigate finitely additive functions with the help of complemented elements of a semiring. We also generalize some classical…
A natural connection between rational functions of several real or complex variables, and subspace collections is explored. A new class of function, superfunctions, are introduced which are the counterpart to functions at the level of…
We develop the foundations of effective ultraproducts of fields and their Galois groups using the methods of computability theory. These computability-theoretic analogs of ultraproducts are called cohesive products, since the role of an…
Counting functions are constructed for sums of integers raised to a fixed positive rational power. That is, given values formed by $u_1^{j/k} + u_2^{j/k} + ... + u_l^{j/k}$, $u_i \in \mathbb{Z}^+$, the number of values less than or equal to…
Observing a multiple version of the divisor function we introduce a new zeta function which we call a multiple finite Riemann zeta function. We utilize some $q$-series identity for proving the zeta function has an Euler product and then,…
In this text we develop the formalism of products and powers of linear codes under componentwise multiplication. As an expanded version of the author's talk at AGCT-14, focus is put mostly on basic properties and descriptive statements that…
We count the number of irreducible polynomials in several variables of a given degree over a finite field. The results are expressed in terms of a generating series, an exact formula and an asymptotic approximation. We also consider the…
One of the fundamental results in computability is the existence of well-defined functions that cannot be computed. In this paper we study the effects of data representation on computability; we show that, while for each possible way of…
We consider binomial and inverse binomial sums at infinity and rewrite them in terms of a small set of constants, such as powers of $\pi$ or $\log(2)$. In order to perform these simplifications, we view the series as specializations of…
A new summation method is introduced to convert a relatively wide family of infinite sums and local expansions into integrals. The integral representations yield global information such as analytic continuability, position of singularities,…
Special functions have always played a central role in physics and in mathematics, arising as solutions of particular differential equations, or integrals, during the study of particular important physical models and theories in Quantum…