Timeline for Bateman-Horn, continued even further

6 events

when toggle format	what		by	license	comment
Sep 1, 2015 at 10:42	comment	added	Igor Rivin		Ah, I see, thanks! If you want to chat off-line, perhaps you could drop me an email ([email protected])?
Sep 1, 2015 at 10:01	comment	added	kantelope		On the other hand, one can calculate these (partial) $L$-values to any precision desired, viewing it as a Dirichlet series (rather than an Euler product) and using an approximate functional equation with about $\sqrt N$ terms, as sketched in Cohen's Section 2 (and gone over in more detail and generality in 10.3 of his second GTM book).
Sep 1, 2015 at 9:56	comment	added	kantelope		In Kurokawa's Theorem A2, he writes $C(f)=\prod_{p\le M}(...)\times \prod_{(\rho,n)\neq (Id,1)} L^M(n,E,\rho)^{...}$, where the first part is an Euler product up to $M$ as wanted, and the second (as in Cohen's writeup) is a product over (partial) $L$-function values, with only $p\ge M$ considered in their Euler products. These partial $L$-functions converge sufficiently rapidly to 1 for $n\gt 1$. However, for $n=1$ (and all nontriv irreps of the Galois group), we again need to estimate how fast the $p\ge M$ part of the Euler product converges to $L^M(1)$, and involves zero-free regions I fear.
Sep 1, 2015 at 9:09	comment	added	Igor Rivin		Thanks! There is another form of the remainder term in Kurokawa's paper - I assume that does not make things easier?!
Sep 1, 2015 at 3:13	history	edited	kantelope	CC BY-SA 3.0	added 134 characters in body
Sep 1, 2015 at 1:23	history	answered	kantelope	CC BY-SA 3.0