Who proved the Prime Number Theorem?
Okay, I think I've asked this question before but I just rummaged through the old topics and cannot find it.
A) Who proved the Prime Number Theorem?
B) Where can I find the proof (on the internet)?
C) Has anyone proven that pi(n) is always MORE than what the Prime Number Theorem would suggest? I.e. has it been proven that the number of primes less than n is always MORE than n/ln(n)?
Both Google and Wikipedia are your friends. I suggest that you get better acquainted with them.
During the 20th century, the theorem of Hadamard and de la Vallée-Poussin also became known as the Prime Number Theorem. Several different proofs of it were found, including the "elementary" proofs of Atle Selberg and Paul Erdős (1949). While the original proofs of Hadamard and de la Vallée-Poussin are long and elaborate, and later proofs have introduced various simplifications through the use of Tauberian theorems but remained difficult to digest, a short proof was discovered in 1980 by American mathematician Donald J. Newman. Newman's proof is arguably the simplest known proof of the theorem, although it is non-elementary in the sense that it uses Cauchy's integral theorem from complex analysis.
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Reading between the lines here. I would assume that these "two proofs of the asymptotic law" would indeed prove that pi(n) is always higher than n/ln(n). Otherwise, there would be at least one point where the two equations pi(n) and n/ln(n) would be identical at some finite point (if viewed continuously/analogically), which by definition breaks down the whole idea of asymptosis. RIGHT?
Extending the ideas of Riemann, two proofs of the asymptotic law of the distribution of prime numbers were obtained independently by Jacques Hadamard and Charles Jean de la Vallée-Poussin and appeared in the same year (1896). Both proofs used methods from complex analysis, establishing as a main step of the proof that the Riemann zeta function ζ(s) is non-zero for all complex values of the variable s that have the form s = 1 + it with t > 0.
Thelibrarian
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Jew, I am not a mathematician by anybody's definition of that word. But I am a librarian and have a book recommendation if you are interested:
http://www.amazon.com/Prime-Obsession-B ... derbyshire
Despite Derbyshire's fall from PC grace, he still has one of the sharpest minds I have ever dealt with.
A) Who proved the Prime Number Theorem?
B) Where can I find the proof (on the internet)?
C) Has anyone proven that pi(n) is always MORE than what the Prime Number Theorem would suggest? I.e. has it been proven that the number of primes less than n is always MORE than n/ln(n)?
Have a look at
http://www.math.sunysb.edu/~moira/mat33 ... 0Prime.pdf
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