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I read an article by Andrew Granville on the subject, there's actually quite a bit of recent literature on the topic. My problem is as follows. I have two sequences of primes: $(p_{1,n})$ and $(p_{3,n})$, respectively primes of the form $4k+1$ and $4k+3$. Because all odd squares are also of the form $4k+1$, there is a bias (Chebyshev's bias): the race is not fair.

Anyway, I am wondering if there are some known bounds on the error term $p_{1,n}-p_{3,n}$. If I read and interpreted correctly, it is reasonable to expect that $|p_{1,n}-p_{3,n}|/ \sqrt{n\log\log n}$ is bounded? I could not find a reference about it. Is it a proven fact? This is the law of the iterated logarithm, I would assume it could be plausible if the race is fair, but this is not the case here (well it is fair but only asymptotically). Do you know the exact statement (or the correct version if mine is wrong) and whether it is unconditional to GRH? I am definitely looking for an unconditional result even if much weaker, especially unconditional to the conjecture that $L(s,\chi_4)$ has no root if $\Re(s)>\frac{1}{2}$.

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    $\begingroup$ The usually studied quantities are $\pi(x;q,a):=\#\{p \le x: p \equiv a \bmod q\}$ so that $p_{i,n}$ ($i\in \{1,3\}$) is the smallest number with $\pi(p_{i,n};4,i)=n$. Under GRH, $(*)\, \pi(x;4,1)-\pi(x;4,3)= O(\sqrt{x}\log x)$ (see Corollary 13.8 in Montgomery and Vaughan's book). It is known that this difference is $\Omega(\sqrt{x}(\log x)^{-1}\log \log \log x)$ (p. 481 of said book) and it is conjectured that the difference is $O(\sqrt{x}(\log x)^{-1}(\log \log \log x)^2)$ and that this is best possible (see p. 484 of said book). All of this follows from the explicit formula. $\endgroup$
    – Ofir Gorodetsky
    Commented Jan 22, 2023 at 1:35
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    $\begingroup$ PNT in arithmetic progressions shows $p_{i,n} \sim 2n\log n$ and with a bit of work, $(*)$ (or rather GRH in the form $\pi(x;4,i) = \mathrm{Li}(x)/2 + O(\sqrt{x} \log x)$) implies that $p_{1,n}-p_{3,n} = O(\sqrt{n}(\log n)^{5/2})$, if I didn't mess the power of log. Using conjectural bounds for $\pi(x;4,i)-\mathrm{Li}(x)/2$ one can do a bit better. The conjecture from the previous comment is related to probabilistic models for $(\sum_{\rho} x^{\rho}/\rho)/\sqrt{x}$, whose maximum one wants to understand. See the literature referred to in p. 484 (Monach and Montgomery). $\endgroup$
    – Ofir Gorodetsky
    Commented Jan 22, 2023 at 1:39
  • $\begingroup$ Thank you Ofir. If I you use $O((n \log n)^{r+\epsilon})$ for the bound, it seems you can prove this weak version of GRH, if my reasoning is correct: no root for $L(s,\chi_4)$ if $\Re(s)>r$. I want to avoid GRH. I assume getting the strength $r=\frac{1}{2}$ without GRH is hopeless, so looking for something weaker. Maybe $r=3/4$, not using GRH. $\endgroup$
    – Vincent Granville
    Commented Jan 22, 2023 at 2:51

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