The Legendre Approximation and Arithmetic Bias

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.