The Legendre Approximation and Arithmetic Bias

TL;DR

This paper explores the historical and mathematical significance of Legendre's prime counting formula, highlighting the role of arithmetic bias and analyzing the Legendre constant's properties and origins.

Contribution

It reveals how arithmetic bias influenced Legendre's formula and introduces a natural criterion related to the Legendre constant, offering insights into its historical development.

Findings

Arithmetic bias likely influenced Legendre's formula.

The Legendre constant 1.08366 satisfies a natural criterion.

Conjecture on how Legendre arrived at the erroneous constant.

Abstract

An interesting episode in the history of the prime number theorem concerns a formula proposed by Legendre for counting the primes below a given bound. We point out that arithmetic bias likely played an important role in arriving at that formula and in its subsequent widespread, decades-long recognition. We also show that the Legendre constant 1.08366 satisfies a certain simple and natural criterion, and conjecture that this criterion is how Legendre arrived at that erroneous constant in his formula.