I'm interested in knowing if there exists a bound (possibly similar to the one found by Baker, Harman, and Pintz) for gaps between primes in arithmetic progression given a fixed quadratic character.

For example, does there exist an $\epsilon > 0$ depending on $d$ and $m$ such that, whenever $\gcd(d,m) = 1$, $\chi$ is the quadratic character of conductor $d$, and $x$ is sufficiently large, there is a prime $p$ congruent to $a \pmod m$ such that $x < p < x + x^{1-\epsilon}$ and $\chi(p) = 1$?

  • $\begingroup$ What if $d=m$, and $\chi(a)=-1$? $\endgroup$ – Gerry Myerson Jun 27 '17 at 6:49
  • $\begingroup$ Changed the gcd so that this problem doesn't occur. $\endgroup$ – The Traveling Salesman Jun 27 '17 at 14:48
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    $\begingroup$ The proof of Hoheisel's theorem (which Baker, Harman and Pintz refine to a better exponent), extends to primes in arithmetic progressions. This was first done by Heilbronn in Uber den Primzahlsatz von Herrn Hoheisel (Math Zeitschrift, 1933). For more recent references and further improvements you may consult pages 141 and 142 of Narkiewicz's Rational Number Theory in the 20th Century. $\endgroup$ – Vesselin Dimitrov Jun 27 '17 at 15:34
  • $\begingroup$ Is there any version of Heilbronn's paper that is in English? $\endgroup$ – The Traveling Salesman Jun 29 '17 at 14:38

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