View Single Post
Old 2018-12-28, 00:09   #1
carpetpool
 
carpetpool's Avatar
 
"Sam"
Nov 2016

23·41 Posts
Post 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/(p-1) - 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 p-th roots of unity (the p-th 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.
carpetpool is offline   Reply With Quote