Timeline for Prime races à la Mertens
Current License: CC BY-SA 3.0
5 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 24, 2014 at 17:36 | comment | added | Jan-Christoph Schlage-Puchta | @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. | |
| Apr 22, 2014 at 19:42 | comment | added | Joël | @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). | |
| Apr 22, 2014 at 18:38 | comment | added | The Masked Avenger | 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. | |
| Apr 22, 2014 at 17:09 | comment | added | Joël | 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... | |
| Apr 22, 2014 at 16:29 | history | answered | Peter Humphries | CC BY-SA 3.0 |