Determining Mills' Constant and a Note on Honaker's Problem

TL;DR

This paper investigates Mills' constant under the Riemann Hypothesis, providing a highly precise calculation of its value and exploring related prime distribution problems, including Honaker's problem.

Contribution

It establishes the value of Mills' constant under RH with extreme precision and discusses prime distribution in short intervals, linking to Honaker's problem.

Findings

Mills' constant begins with 1.3063778838 under RH

Calculated Mills' constant to 6850 decimal places

Applied Cramér-Granville Conjecture to Honaker's problem

Abstract

In 1947 Mills proved that there exists a constant $A$ such that $\lfloor A^{3^n} \rfloor$ is a prime for every positive integer $n$. Determining $A$ requires determining an effective Hoheisel type result on the primes in short intervals - though most books ignore this difficulty. Under the Riemann Hypothesis, we show that there exists at least one prime between every pair of consecutive cubes and determine (given RH) that the least possible value of Mills' constant $A$ does begin with 1.3063778838. We calculate this value to 6850 decimal places by determining the associated primes to over 6000 digits and probable primes (PRPs) to over 60000 digits. We also apply the Cram\'er-Granville Conjecture to Honaker's problem in a related context.