Are primes equally likely to be equivalent to $1$ or $3$ modulo $4,$ or is there a skew in one direction?
That is my specific question, but I would be interested to know if there exists a trend more generally, say for modulo any even.
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2$\begingroup$ I’d recommend you read JOURNAL ARTICLE Prime Number Races (though I found an error: 93 is not prime) Andrew Granville and Greg Martin The American Mathematical Monthly Vol. 113, No. 1 (Jan., 2006), pp. 1-33 $\endgroup$– J. W. TannerApr 22, 2019 at 13:58
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1$\begingroup$ Do I understand correctly that your pool is from 3 up to some greater prime $p_k$ (that is to say, $k > 2$)? Then it depends on what $p_k$ is. Define a function $f(p) = -1$ if $p \equiv 3 \pmod 4$, otherwise $f(p) = 1$. Then I would hazard a guess that $$\sum_{i = 2}^k f(p)$$ changes sign infinitely often. $\endgroup$– Robert SoupeApr 22, 2019 at 16:25
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2 Answers
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In general, if $\gcd(a,b) = 1$, the number of primes which are of the form $b$ modulo $a$ is asymptotic to $\dfrac{\pi(x)}{\varphi(a)}$ where $\pi(x)$ is the number of primes $\le x$. As you see the asymptotic formula is independent of $b$ hence for a given modulo all residues occur equally in the long run.
answered Apr 22, 2019 at 12:54
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Despite of Chebychev's bias (more primes with residue $3$ occur usually in practice), in the long run, the ratio is $1:1$.
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1$\begingroup$ I would also note for future reference that the proof is given as an immediate consequence of Dirichlet’s Theorem and in general the primes are equidistributed modulo every $d$ with approximately $\frac{1}{\phi(d)}$ in each class. $\endgroup$ Apr 22, 2019 at 12:51
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2$\begingroup$ Generalization of Chebyshev's biase : The primes with a quadratic non-residue usually occur more often in practice than the primes with a quadratic residue. $\endgroup$– PeterApr 22, 2019 at 12:54
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$\begingroup$ Interesting I haven't heard of that generalization before, is there any reason other than empirical evidence that this happens? $\endgroup$ Apr 22, 2019 at 12:57
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$\begingroup$ I only know that Chebyshev noticed this phenomenon, no idea whether it has a mathematical reason. Also interesting : In the "race" between primes of the form $4k+1$ and $4k+3$ it is known that the lead switches infinite many often. $\endgroup$– PeterApr 22, 2019 at 13:00