non-asymptotic Bertrand-type theorems for arithmetic progression - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T16:58:07Z http://mathoverflow.net/feeds/question/55355 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/55355/non-asymptotic-bertrand-type-theorems-for-arithmetic-progression non-asymptotic Bertrand-type theorems for arithmetic progression Fedor Petrov 2011-02-13T23:24:07Z 2011-02-18T18:54:25Z <p>It is well known that primes of form $4k+3$, call them $3=q_1 &lt; q_2 &lt; \dots$ satisfy $q_{n+1}/q_n\rightarrow 1$ (and even $q_n=\frac{n}{2\log n}(1+o(1))$). I would be glad to see results of Dusart-type (http://mathoverflow.net/questions/2724/strong-bertrand-postulate/2729#2729) with concrete bounds for $N$ such that $q_{n+1} &lt; 1.1 q_n$ provided $q_n > N$ or something like this (I am not sure that I need a multiple exactly $1.1$)</p> http://mathoverflow.net/questions/55355/non-asymptotic-bertrand-type-theorems-for-arithmetic-progression/55369#55369 Answer by Felipe Voloch for non-asymptotic Bertrand-type theorems for arithmetic progression Felipe Voloch 2011-02-14T00:38:58Z 2011-02-14T00:38:58Z <p>Have a look at (Satz 9 and the preceeding calculations):</p> <p>1935-10 P. Erdõs: Über die Primzahlen gewisser arithmetischer Reihen (in German), Math. Z. 39 (1935), 473--491; Zentralblatt 10,293. </p> <p>Available at: <a href="http://www.renyi.hu/~p_erdos/Erdos.html" rel="nofollow">http://www.renyi.hu/~p_erdos/Erdos.html</a></p>