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Using some suitable L-series for some appropriate ray class group, one can find the Dirichlet density of some set of primes. One can conclude that this set of prime is infinite as long as the density is non-zero. I wonder if this method has been tried for, e.g. regular primes. Perhaps it's not easy to connect the in-divisibility with the Frobenius elements? What's so impractical about it for regular primes? (Feel free to close it if this question is too vague of too 'big')

edited Oct 31 2010 at 23:29

asked Oct 31 2010 at 23:24

Jizhan Hong
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Whether or not a prime is regular isn't determined by congruence conditions, so it's hard to see how you could build a Dirichlet series with an Euler product expansion out of them. Dirichlet's work (and the generalization you cite to ray class groups) depends crucially on the multiplicativity of characters. – Hunter Brooks Nov 1 2010 at 0:55

Hunter: This may be nonsense, but one can start with this: suppose towards a possible contradiction that the set of regular primes is finite, then let $m$ be the product of the regular primes, consider the modulus $(m)p_\infty$ on $\mathbf{Q}$, then.... I don't know what happens, but maybe one can get a different density for some set of primes, and hence a contradiction? – Jizhan Hong Nov 1 2010 at 13:33

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Jizhan, what you write in your comment is nonsense as far as L-series methods are concerned, since taking the product of the primes sounds more like Euclid's proof than a proof using L-series (Dirichlet's theorem on primes doesn't use a product of primes satisfying a congruence condition). Siegel conjectured that the set of regular primes has natural density 1/sqrt(e) (see the wikipedia page on regular primes), and it seems pretty doubtful that L-series techniques would explain a density like that. – KConrad Nov 2 2010 at 4:28

KConrad: I guess you are right. But, note that I used the word 'modulus', implicitly suggesting that one can construct a ray class out of it, and then a Dirichlet character and some L-series. And maybe one can implement Frobenius' Density Theorem somehow; I was just suggesting to Hunter Brooks that one can still get started somehow. I knew about Siegel's conjecture before asking the question, but I was/am only concerned about the 'infiniteness' part of the original conjecture. And I knew that even for this part, it's probably hopeless to use L-series. – Jizhan Hong Nov 2 2010 at 13:26

If an L-series is going to help at all it probably is going to spit out a density formula even if you aren't asking it to, and that's why the "strange" conjectural density 1/sqrt(e) suggests L-series are not likely to help. If you are only concerned with the infinitude, then it would sound better just to ask how, given a list of regular primes, one can find a new one. Of course nobody knows, which is why the problem is still open. Elkies showed ell. curves over Q have inf. many supersingular primes by a recursive approach, not by L-series. – KConrad Nov 2 2010 at 23:25

L-series method, how far can it go?

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