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

Eugenio P. Balanzario

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.

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

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.

Paper Structure

This paper contains 3 sections, 3 theorems, 45 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Proof of Theorem t2
  3. Proof of Theorem t1

Key Result

Theorem 1

Assume the Riemann hypotheses and the simplicity of the zeros $\beta+i\gamma$ of the zeta function. Let $\sigma$, $\eta$ and $\theta$ be positive real numbers. For $n\in\mathbb{N}$, let $S_\sigma(n)$ be as in (sden) and for $x\in\mathbb{R}$, let Then we have that, as $n\to\infty$,

Figures (3)

  • Figure 1: Illustrates formula (\ref{['sdex']}) of Theorem \ref{['t1']} by showing $\widehat{S}_\sigma(x)$ for $x\in(a,b)$ with $b=10^9$ and $a=0.5 b$. Notice that $\widehat{S}_\sigma(x)$ oscillates around its expected value 1. With dashed lines are marked the levels $1\pm2\sigma$, so that approximately $95.45\%$ of the values of $\widehat{S}_\sigma(x)$ lie within these limits.
  • Figure 2: Illustrates Theorem \ref{['t2']} by showing one hundred uniformly selected values of $\widehat{S}_\sigma(x)$ for each one of the following four values of $b$. In part (a) of the figure we have set $b=1.25\times10^8$, in part (b) we have $b=2.5\times10^8$, in part (c) we have $b=5\times10^8$ and in part (d) we have $b=10^9$. In each part of this figure, in continuous line is the corresponding distribution of Theorem \ref{['t2']}.
  • Figure 3: In part (a) of the figure, we present the dispersion diagram of 2 005 points of the form $(\widehat{S}_\sigma(x), R_\sigma(x))$ where $R_\sigma(x)$ is as in equation (\ref{['erre']}). The values of $\widehat{S}_\sigma(x)$ in the abscissa are those that appear in Figure \ref{['fig1']} with $b=10^9$ and $a=0.5 b$. In part (b) of the figure is the histogram of the ordinates $R_\sigma(x)$ in the previous dispersion diagram. In dashed line is the density function of a normal probability distribution with mean 1 and variance as in formula (\ref{['varianza']}).

Theorems & Definitions (4)

  • Theorem 1
  • Theorem 2
  • Lemma 1
  • proof