Related papers: Various Arithmetic Functions and their Application…
We introduce a notion of complexity of diagrams (and in particular of objects and morphisms) in an arbitrary category, as well as a notion of complexity of functors between categories equipped with complexity functions. We discuss several…
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…
This thesis determines some of the implications of non-universal and emergent universal statistics on arithmetic correlations and fluctuations of arithmetic functions, in particular correlations amongst prime numbers and the variance of the…
We prove that several results in different areas of number theory such as the divergent series, summation of arithmetic functions, uniform distribution modulo one and summation over prime numbers which are currently considered to be…
We connect Dedekind sums and some formulas for numerical semigroups.
Motivated by the problem of finding finite versions of classical incompleteness theorems, we present some conjectures that go beyond ${\bf NP\neq co NP}$. These conjectures formally connect computational complexity with the difficulty of…
We define an enumerative function F(n,k,P,m) which is a generalization of binomial coefficients. Special cases of this function are also power function, factorials, rising factorials and falling factorials. The first section of the paper is…
Consider the representation of a rational number in the form, associated with "centered" Euclidean algorithm. We prove a new formula for the limit distribution function for sequences of rationals with bounded sum of partial quotients.
Fracture functions and their evolution equations are reviewed. Some phenomenological applications are briefly discussed.
This lecture addresses some general ideas behind numerical computations ranging from representation of numbers in computers to stability and accuracy of standard algorithms for some simple mathematical problems.
The paper is devoted to the implicit function theorem involving singular mappings.We also discuss the form of the tangent cone to the solution set of the generalized equations in singular case and give some examples of applications to…
It is quite well-known from Kurt Godel's (1931) ground-breaking result on the Incompleteness Theorem that rudimentary relations (i.e., those definable by bounded formulae) are primitive recursive, and that primitive recursive functions are…
We investigate the computational properties of basic mathematical notions pertaining to $\mathbb{R}\rightarrow \mathbb{R}$-functions and subsets of $\mathbb{R}$, like finiteness, countability, (absolute) continuity, bounded variation,…
Let $p$ be an odd prime. In the paper we collect the author's various conjectures on congruences modulo $p$ or $p^2$, which are concerned with sums of binomial coefficients, Lucas sequences, power residues and special binary quadratic…
We consider a family of integer sequences generated by nonlinear recurrences of the second order, which have the curious property that the terms of the sequence, and integer multiples of the ratios of successive terms (which are also…
We will see that key concepts of number theory can be defined for arbitrary operations. We give a generalized distributivity for hyperoperations (usual arithmetic operations and operations going beyond exponentiation) and a generalization…
We employ the notions of `sequential function' and `interrogation' (dialogue) in order to define new partial combinatory algebra structures on sets of functions. These structures are analyzed using J. Longley's preorder-enriched category of…
Let $p$ be a prime. We discuss $p$-adic properties of various arithmetical functions related to the coefficients of modular form and generating functions. Modular forms are considered as a tool of solving arithmetical problems. Examples of…
In [arXiv:2212.04969], the authors stated some conjectures on the variance of certain sums of the divisor function $d_k(n)$ over number fields, which were inspired by analogous results over function fields proven in [arXiv:2107.01437].…
In this note we list a number of open problems in the fields of number theory, combinatorics, and representation theory: algebraic functions with Fermat property; power product expansion of the generating function for the partition…