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A quick look at the primes in $\mathbb{Z}[i]$ suggests they might be evenly distributed by angle if we zoom out on a coarse enough scale.

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I would like ask about the much weaker statement forgetting questions about angular bias.

Do all sectors $(\theta, \phi)$ have infinitely many primes $ \mathfrak{p} $?

The idea being to find cases where it is possible to mimick Euclid's proof of the infinitude of primes in $\mathbb{Z}$. I was not able to find such an elementary proof. Certainly let $|\theta - \phi| \ll \frac{\pi}{8}$.

This may be like the cause of Dirichlet's theorem of primes in arithmetic progressions, Euclid's proof extends to some arithmetic progressions (like $4k+3,6k+5, 8n+5$) and not others [1]

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    $\begingroup$ Indeed the angular equidistribution of the Gaussian primes is completely similar to Dirichlet's theorem (on the logarithmic equidistribution of the primes in an arithmetic progressions). Either is equivalent to the non-vanishing at $s = 1$ of a certain Hecke (resp. Dirichlet) $L$-function. If you rather look for cases where a Euclid argument applies, Elkies's theorem must be mentioned: every elliptic curve over $\mathbb{Q}$ has infinitely many supersingular primes. $\endgroup$
    – Vesselin Dimitrov
    Commented Jul 25, 2015 at 14:40
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    $\begingroup$ You are well aware of the equidistribution of Gaussian primes by angle, as your previous question testifies (mathoverflow.net/questions/206707/…). So it is not clear what you are asking. The "cause of a theorem" makes no sense: we are dealing here with mathematics, not theology (in the words of André Weil). At any rate, Dirichlet's theorem and the quoted equidistribution theorem of Hecke have a common source, the nonvanishing of certain $L$-values. $\endgroup$
    – GH from MO
    Commented Jul 25, 2015 at 15:30
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    $\begingroup$ It's a complexity issue, right? Non vanishing of L-functions is a lot of "work" to prove involving complex analysis and such. The hope is I can prove weaker statements in the same direction with less effort. Then what is the best I can do? $\endgroup$
    – john mangual
    Commented Jul 25, 2015 at 15:50
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    $\begingroup$ So, it seems that the question you are actually asking is something like, "Is there an elementary proof that all sectors have infinitely many primes?" If so, you might want to edit the question to make that clearer. $\endgroup$
    – Gerry Myerson
    Commented Jul 25, 2015 at 23:44
  • $\begingroup$ Ah. What I had in mind was an edit to the body of the question (not so much the title). $\endgroup$
    – Gerry Myerson
    Commented Jul 26, 2015 at 6:10

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Yes, and in fact there will still be many primes even if the size of the sector decrease to zero quite quickly.

In the 2001 paper "Gaussian primes in narrow sectors," Harman and Lewis use sieve methods to prove that there are Gaussian primes with $|p|^2\leq X$ in sectors where the angle goes to zero at the rate of $X^{-0.381}$. Specifically they prove that:

Theorem: Let $X>X_0$. Then the number of Gaussian primes $p$ with $|p|^2\leq X$ in the sector $\beta\leq \arg p\leq \beta +\gamma$ is at least $$\frac{cX\gamma}{\log X}$$ for an absolute constant $c>0$, where $\beta,\gamma$ satisfy $0\leq \beta \leq \frac{\pi}{2},$ $X^{-0.381}\leq \gamma \leq \frac{\pi}{2}$.

One interesting corollary of this result is that there are infinitely many primes $p$ satisfying $$\{\sqrt{p}\}<p^{-0.262},$$ where $\{x\}$ denotes the fractional part of $x$.

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