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I have just read the nice survey by Granville and Martin about prime races. I wonder what happens if one changes the rules for the prime races as follows. Fix $q$ a modulus (an integer $>1$). For $a$ an integer relatively prime to $q$, one has a Mertens-like formula $$\sum_{p<x, \atop p \equiv a \pmod{q}} \frac{1}{p} = \frac{1}{\phi(q)} \log\log x + M_{q,a} + O(1/\log x)$$ where $M_{q,a}$ is a constant. We can say that the "team $a$" wins the race mod $q$ if the constant $M_{q,a}$ is greater than the constants $M_{a,b}$ for others $b \pmod{q}$, and we can even make a complete ranking of the various teams (with possible ties). This is a reasonable notion, since stating that $M_{q,a} > M_{q,b}$ implies that the number of primes $p$ up to $x$ counted with harmonic density $1/p$ which are congruent to $a$ mod $q$ is greater than the same number for congruent to $b$ mod $q$, for every $x$ large enough.

So what can be said about the ranking of the $M_{q,a}$'s for various $a$ mod $q$?

For example, with sage I have computed approximation of the values of $M_{8,a}$ for $a=1,3,5,7$, and I find numbers close to $-0.28,0.16,0.00,-0.11$ making the "team 3" the clear winner.

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I think this paper gives a pretty good answer: http://projecteuclid.org/euclid.em/1317758092

In the appendix, Languasco and Zaccagnini give a proof of Norton that if $q \geq 2$ and $1 \leq a < q$ with $(q,a) = 1$, then as $q \to \infty$, $$M_{q,a} = \begin{cases} \displaystyle \frac{1}{a} + O\left(\frac{\log q}{\varphi(q)}\right) & \text{if $a$ is prime,} \\ \displaystyle O\left(\frac{\log q}{\varphi(q)}\right) & \text{otherwise.} \end{cases}$$ This tells you that the race ought to be won by the smallest prime coprime to $q$, at least for $q$ sufficiently large. If you race two nonprimes against each other, however, then this says nothing.

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  • $\begingroup$ This is interesting, and surprising. Being a quadratic non residue seems to play no role in those Mertens rate, while being a prime between $1$ and $q$ does... $\endgroup$
    – Joël
    Commented Apr 22, 2014 at 17:09
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    $\begingroup$ It does not seem surprising to me. Considering how slowly the race is run (slow growth), having a big early start (reciprocal of small prime) should ensure success. This is evidenced by the mod 8 results. $\endgroup$
    – The Masked Avenger
    Commented Apr 22, 2014 at 18:38
  • $\begingroup$ @TheMaskedAvenger: the race is slow but still it goes to infinity, so the advantage given by a good start shall eventually disappear. What I find surprising is how different the winners and loser look from the case of usual prime races (where winners are non-residues, loser residues). $\endgroup$
    – Joël
    Commented Apr 22, 2014 at 19:42
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    $\begingroup$ @Joel: When passing from $\Psi$ to $\theta$, you introduce a systematic bias of magnitude $c_q\sqrt{x}$ favouring non-squares over squares. The zeros of the relevant $L$-series introduce a "random" term of the same magnitude, so when considering $\pi(x,q,a)$ we expect that most of the times the non-squares lead, but sometimes the squares overtake for a short time. But when considering $\sum\frac{1}{p}$, partial summation shows that this bias becomes $\mathcal{O}(q^{-1/2+\epsilon})$, which is negligible compared to the contribution of the small primes. $\endgroup$
    – Jan-Christoph Schlage-Puchta
    Commented Apr 24, 2014 at 17:36

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