On one number-theoretic conception: towards a new theory

TL;DR

This paper introduces a novel number-theoretic framework based on a new ordering of integers and an axiom system, providing a unified approach to summation of divergent series and deriving new formulas for Bernoulli numbers.

Contribution

It proposes a new method for ordering integers, a new axiom system, and a regular summation method for divergent series, offering fresh insights and simpler derivations of known results.

Findings

Unified approach to summation of divergent series

Explicit formulas for sums of some infinite series

New recurrence formulas for Bernoulli numbers

Abstract

In this paper we present a new mathematical conception based on a new method for ordering the integers. The method relies on the assumption that negative numbers are beyond infinity, which goes back to Wallis and Euler. We also present a new axiom system, the model of which is arithmetics. We define regular method for summation of infinite series which allows us to discover general and unified approach to summation of divergent series, and determine the limits of unbounded and oscillating functions. Several properties for divergent series and explicit formulas for sums of some infinite series are established. A number of finite and new recurrence formulas for Bernoulli numbers are obtained. We rederive some known results, but in a simpler and elementary way, and establish new results by means of techniques of the theoretical background developed.