Prime Number Theorem w/o Complex Analysis - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T15:01:52Z http://mathoverflow.net/feeds/question/60607 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/60607/prime-number-theorem-w-o-complex-analysis Prime Number Theorem w/o Complex Analysis LowerBounds 2011-04-04T20:50:21Z 2011-07-08T17:36:12Z <p>I know about</p> <p>"Simple analytic proof of the prime number theorem" Newman, 1980</p> <p>However, is there a proof of the Prime Number Theorem without the use of complex analysis? (Real analysis is fine).</p> <p>Thanks!</p> http://mathoverflow.net/questions/60607/prime-number-theorem-w-o-complex-analysis/60611#60611 Answer by Charles Matthews for Prime Number Theorem w/o Complex Analysis Charles Matthews 2011-04-04T21:12:58Z 2011-04-04T21:12:58Z <p><a href="http://www.math.columbia.edu/~goldfeld/ErdosSelbergDispute.pdf" rel="nofollow">http://www.math.columbia.edu/~goldfeld/ErdosSelbergDispute.pdf</a> explains the classic proof in context (there is what amounts to a priority dispute).</p> http://mathoverflow.net/questions/60607/prime-number-theorem-w-o-complex-analysis/60622#60622 Answer by Gerry Myerson for Prime Number Theorem w/o Complex Analysis Gerry Myerson 2011-04-05T00:06:39Z 2011-04-05T00:06:39Z <p>Another exposition of an elementary proof (that is, a proof not using complex analysis) is in Gerald Tenenbaum and Michel Mendes France, The Prime Numbers and Their Distribution, which is Volume 6 of the Student Mathematical Library, published by the American Mathematical Society. The proof they give is due to Daboussi, from 1984. </p> http://mathoverflow.net/questions/60607/prime-number-theorem-w-o-complex-analysis/60840#60840 Answer by M Mueger for Prime Number Theorem w/o Complex Analysis M Mueger 2011-04-06T17:38:38Z 2011-04-06T17:38:38Z <p>A nice exposition of an Erdos/Selberg-type elementary proof is given by Levinson in Amer. Math. Monthly 76 (1969) 225–245.</p> <p>The proof by Daboussi as written up by Tenenbaum and Mendes-France was already mentioned.</p> <p>Yet another one is due to Hildebrand in Mathematika 33 (1986) 23–30. </p> http://mathoverflow.net/questions/60607/prime-number-theorem-w-o-complex-analysis/60844#60844 Answer by Emerton for Prime Number Theorem w/o Complex Analysis Emerton 2011-04-06T18:51:30Z 2011-04-06T18:51:30Z <p>There is a terrific exposition of the elementary proof by Terry Tao, available as the file prime.dvi <a href="http://www.math.ucla.edu/~tao/preprints/Expository/" rel="nofollow">here</a>. A more traditional exposition is available in Edwards's book <em>Riemann's zeta function</em>.</p> http://mathoverflow.net/questions/60607/prime-number-theorem-w-o-complex-analysis/69809#69809 Answer by Richard Dore for Prime Number Theorem w/o Complex Analysis Richard Dore 2011-07-08T17:36:12Z 2011-07-08T17:36:12Z <p>If you just want $\pi(n) = \Omega \left( \frac{n}{\log n} \right)$, good enough for many applications, here is a quick proof: The highest power of a prime $p$ dividing $2n \choose n$ is at most $2n$ -- you get at most one more factor of $p$ in the numerator than denominator for each power $p^i \leq 2n$. This tells you that ${2n \choose n} \leq (2n)^{\pi(2n)}$. So $\pi(2n) \geq \frac{\log_2 {2n \choose n}}{\log_2 (2n)} \geq \frac{n}{\log_2 (2n)}$.</p>