20181228, 00:09  #1 
"Sam"
Nov 2016
517_{8} Posts 
Density of norms (field theory)
In this post here, I asked for the conditional probability for an integer N being prime given that all prime q dividing n are congruent to 1 modulo 2*p (for some prime p). As a result, I also got the answer of how many integers N not exceeding x can be written as a product of primes only congruent to 1 modulo 2*p. This is asymptotically D(x) = c*x*(log(x))^(1/(p1)  1) for some constant c, which seems to be decreasing significantly as p increases.
How many integers N not exceeding x can (I) be written as a product of primes only congruent to 1 modulo 2*p and (II) in addition to (I), N can be expressed as the norm for some integral element f in the ring of integers in K=Q(zeta(p)) where K is the field of pth roots of unity (the pth cyclotomic field) ? The condition for (II) can be restated as there is at least one ideal of norm N that is principal in K. I am hoping for a precise answer (as in my last thread) in an attempt to solve another problem related to this. Again, any information is helpful, and thanks for help. 
20181228, 00:19  #2  
"Forget I exist"
Jul 2009
Dumbassville
2^{4}×3×5^{2}×7 Posts 
Quote:
Last fiddled with by science_man_88 on 20181228 at 00:20 

20181228, 19:05  #3 
"Sam"
Nov 2016
5·67 Posts 
It seems to be that the answer is 0, although I don't know for sure because many of my previous posts seem to be getting lack of attention due to the little known information and research of these topics. However, I did find this article. Still, any other explanations of it are welcome.

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