Performances piecewise defined functions in analytic form, prime-counting function, $\xi$ sets

TL;DR

This paper explores exact analytic representations of discrete functions like the prime-counting function and introduces new $\xi$ sets, providing insights into set theory and mathematical physics.

Contribution

It presents a novel approach to representing discrete functions analytically without approximations and introduces $\xi$ sets inspired by mathematical and physical concepts.

Findings

Exact analytic forms for prime-counting function and piecewise functions

Introduction of $\xi$ sets and their application to Russell's paradox

Insights into set theory and physics through new set concepts

Abstract

The article discusses the representation of discrete functions defined in an analytic form without the use of approximations, namely the Heaviside function, identity function, the Dirac delta function and the prime-counting function. Also in the article introduced and considered a new type of sets ($\xi$ set) by analogy taken from finding the sum of a number of grants and other contradictions in mathematics and physics. With the help of $\xi$ sets interpreted Russell's paradox in the system of axioms of naive set theory.