On the fractal nature of the partition function $p(n)$ and the divisor functions $d_i(n)$

TL;DR

This paper explores the fractal properties of the partition function and divisor functions, presenting new iterative formulas and a network-based representation that reveals their self-similar, fractal nature.

Contribution

It introduces a novel network-based, iterative approach to express partition and divisor functions, highlighting their fractal and self-similar structures.

Findings

Derived an exact iterative expression for partitions of integers.

Introduced the trace function to compute the number of divisors.

Revealed the fractal, self-similar nature of divisor functions.

Abstract

The partitions of the integers can be expressed exactly in an iterative and closed-form expression. This equation is derived from distributing the partitions of a number in a network that locates each partition in a unique and orderly position. From this representation an iterative equation for the function of the number of divisors was derivated. Also, the number of divisors of a integer can be found from a new function called the trace of the number n, trace(n). As a final preliminary result, using the Bressoud-Subbarao theorem, we obtain a new iterative representation of the sum of divisor function. Using this theorem it is possible to derive an iterative equation for any divisor function and all their networks representation will exhibits a self-similarity behavior. We must then conclude that the intricate nature of the divisor functions results from the fractal nature of the partition function described in the present work.