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This question is about Legendre's prime-counting function, the one that can be used to calculate the exact amount of prime numbers that are less than or equal to a given number (as long as the number in question is not too large).

Basically, I'm trying to find references for Legendre's prime-counting function. I searched on the web but I couldn't find many references, except maybe this Wikipedia article and this MathWorld article.

Does anyone have any other references for Legendre's prime-counting function, such as books or journal articles?

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  • $\begingroup$ You mean proof or history? The proof is straightforward, just use the inclusion-exclusion principle. $\endgroup$ – Ian Mateus Jun 15 '14 at 1:19
  • $\begingroup$ @Ian Ok, so what about history? $\endgroup$ – User X Jun 15 '14 at 1:23
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The Wikipedia article you link and its subject matter twin at MathWorld have about 40 unique references between them. I might start there, generically. Note that neither of these websites are "primary sources". If you are searching for primary sources, you will need to find Legendre's "Essai sur la theorie des nombres", 1798. There are many proofs, including those by Lipschitz, Sylvester, Catalan, and Kronecker, among others. (List is abridged from Dickson, "History of the Theory of Numbers", vol. 1, p. 429, 2005.)

The MathWorld page you link has a reference to the excellent Seroul (S\'eroul) text.

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