Chebyshev's approach to the distribution of primes - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T07:00:26Z http://mathoverflow.net/feeds/question/26342 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/26342/chebyshevs-approach-to-the-distribution-of-primes Chebyshev's approach to the distribution of primes Fedor Petrov 2010-05-29T08:38:42Z 2012-09-28T19:04:20Z <p>This is motivated by a recent <a href="http://mathoverflow.net/questions/26336/integer-valued-factorial-ratios" rel="nofollow"> question</a> by Wadim.</p> <p>The negative answer should be known, since t is very natural, in this case I would be happy to see any reference.</p> <p>May Pafnuty Lvovich Chebyshev's approach to distribution of primes lead to PNT itself, if we replace $\frac{(30 n)! n!}{(15 n)! (10 n)! (6 n)!}$ to other integer ratios of factorials? If not, what are the best constants in asymptotic relation $$c_1 \frac{n}{\log n}&lt; \pi(n)&lt; c_2 \frac{n}{\log n}$$ which may be obtained on this way?</p> http://mathoverflow.net/questions/26342/chebyshevs-approach-to-the-distribution-of-primes/26344#26344 Answer by J. H. S. for Chebyshev's approach to the distribution of primes J. H. S. 2010-05-29T08:54:37Z 2012-09-28T19:04:20Z <p>Erdős and Diamond proved in [<strong>1</strong>] that Chebyshev could have achieved sharper bounds for the asymptotic behavior of the prime counting function. Nevertheless, their proof does not shed any light on the first question that you posed because they took the PNT for granted throughout their note.</p> <p>References:</p> <p>[<strong>1</strong>] <strong>H. G. Diamond; P. Erdős.</strong> <em>On sharp elementary prime number estimates</em>, Enseign. Math. (2) 26 (1980) 313-321.</p> http://mathoverflow.net/questions/26342/chebyshevs-approach-to-the-distribution-of-primes/26345#26345 Answer by Robin Chapman for Chebyshev's approach to the distribution of primes Robin Chapman 2010-05-29T09:00:56Z 2010-05-29T09:00:56Z <p>According to the notes in fifth edition of Niven, Zuckerman and Montgomery's <em>An Introduction to the Theory of Numbers</em> for each $\epsilon\in(0,1)$ there is a series of parameters in Chebyshev's method that proves $$(1-\epsilon)\frac{\log x}{x} &lt; \pi(x) &lt; (1+\epsilon)\frac{\log x}{x}$$ for all large enough $x$ but that the proof of this uses PNT so that it doesn't provide an alternative proof of PNT.</p> <p>They cite a paper of H. G. Diamond and P. Erdos: "On sharp elementary prime estimates", <em>L'Enseignment Math.</em> <strong>26</strong> (1980), 313-321.</p>