User mrm - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-18T22:53:10Z http://mathoverflow.net/feeds/user/2166 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/8056/what-are-good-non-english-languages-for-mathematicians-to-know/8093#8093 Answer by mrm for What are good non-English languages for mathematicians to know? mrm 2009-12-07T09:53:10Z 2009-12-07T09:53:10Z <p>French, German, Russian. It's a pity English dominates so much. French reads beautifully. (Also if you read older papers by Hadamard, Stieltjes or Levy, the first thing you'll notice is their extreme honesty (they don't use difficult terms, they define everything, they don't try to make their arguments <em>appear</em> difficult) also calculations are not condemned nor indulged with (they often just write an equation and say in words how the rest of the computation goes)).</p> http://mathoverflow.net/questions/7039/typical-value-of-totient-function/7147#7147 Answer by mrm for Typical value of totient function mrm 2009-11-29T17:49:24Z 2009-11-29T17:49:24Z <p>Let me also mention the following: You can adapt Schoenberg's result to prove that 1/M * {N &lt;= n &lt;= N + M : phi(n) / n &lt;= t} --> F(t) uniformly in t, where F is a distribution function. The proof goes by computing the moments sum((phi(n)/n)^k , N &lt;= n &lt;= N + M). You can probably get a O(loglog N / log N) rate of convergence (as was done by Levin ... if I recall correctly). </p> http://mathoverflow.net/questions/7969/irreducible-polynomials-with-constrained-coefficients/8016#8016 Comment by mrm mrm 2009-12-06T20:19:55Z 2009-12-06T20:19:55Z yes, indeed, i realized that the problem is different only after having posted the message -- my browser doesn't display TeX properly so I made a guess on what the question was... (not a good idea) http://mathoverflow.net/questions/7039/typical-value-of-totient-function/7147#7147 Comment by mrm mrm 2009-11-29T17:59:39Z 2009-11-29T17:59:39Z + If this is of interest the distribution function F(t) decays doubly exponentially at 0, that is F(1/t) &lt;&lt; exp(-C*exp(t)) for some constant C. This was investigated by Erdos and more recently Weingartner (<a href="http://www.mrlonline.org/proc/2007-135-09/S0002-9939-07-08771-0/S0002-9939-07-08771-0.pdf" rel="nofollow">mrlonline.org/proc/2007-135-09/&hellip;</a>). Asymptotics for 1 - F(t) when t is close to 1 where studied by Tenenbaum and Toulmonde (a reference is in the paper above). In this case the asymptotic behaviour is more tame. There should be no problem adapting all these results to the case of the interval [N; N + M] with M as you described...