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arXiv · 2208.02623

The Legendre Approximation and Arithmetic Bias

Ghaith Hiary·Megan Paasche
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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.

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Ghaith Hiary, Megan Paasche. 2022-07-29. The Legendre Approximation and Arithmetic Bias. https://arxiv.org/abs/2208.02623

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