A local to global principle for expected values

Geometric sieve over number fields for higher moments

“…In [ 18 ] the authors generalized Theorem 2.2, the geometric sieve, also called local to global principle, over the integers, to expected values. We will now generalize Theorem 2.4 , the geometric sieve over number fields, to higher moments.…”

“…Still we find it useful to include this possiblity to modifity the box . Even though does no longer correspond to the set of all places, we will keep using the same notation in order to stay consistent with our previous paper [ 18 ]. Note that the set of elements living in infinitely many , i.e., has density zero; this follows directly from Condition ( 2 ).…”

“…As in [ 18 ] we can restrict Definition 3.1 to subsets of , i.e., we define the n -th moment of the system restricted to to be if it exists. We will write for and if it does not depend on the integral basis , we will just write , respectively for .…”

“…For any non-negative integer n , one can easily pass from to and vice versa. The proof is the same as in [ 18 , Lemma 11].…”

“…Recently, a new question regarding densities has arisen in [ 14 ]: to compute the expected value and the variance of Eisenstein polynomials over the integers. More precisely, the question can be formulated as “On an average, for a random Eisenstein polynomial how many primes p are such that this Eisenstein polynomial satisfies the criterion of Eisenstein for p?” This led to [ 18 ], where a general definition of the expected value corresponding to the density is given and an addendum to the geometric sieve over the integers is provided, which allows to compute these expected values corresponding to the density directly using the local characterization of the target set. In addition, in the thesis [ 26 ] the variance of Eisenstein polynomials and other target sets over the integers were determined using an adaption of this new geometric sieve.…”