A note on the distribution of clusters and deserts of prime numbers

TL;DR

This paper investigates the distribution of weighted sums of the von Mangoldt function, providing evidence that they exhibit a distribution similar to a normal distribution, using trigonometric polynomial analysis.

Contribution

It introduces a novel approach to analyze the distribution of weighted sums of the von Mangoldt function through trigonometric polynomial formulas.

Findings

Weighted sums of the von Mangoldt function follow a normal-like distribution

Use of trigonometric polynomial formulas to analyze distribution

Provides evidence supporting normal-like behavior in prime number clusters

Abstract

In this note we consider the distribution of values of weighted sums of the von Mangoldt arithmetical function. By using a formula for the distribution of values of trigonometric polynomials, we are able to present evidence supporting the claim that these weighted sums follow a distribution with a normal-like behavior.