Distribution of associate primes modulo q in number fields - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-26T06:07:58Z http://mathoverflow.net/feeds/question/29258 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/29258/distribution-of-associate-primes-modulo-q-in-number-fields Distribution of associate primes modulo q in number fields Tony 2010-06-23T16:51:04Z 2010-06-23T18:41:05Z <p>In <a href="http://mathoverflow.net/questions/29190/dirichlets-theorem-for-number-fields" rel="nofollow">http://mathoverflow.net/questions/29190/dirichlets-theorem-for-number-fields</a>, I asked about an analogue of Dirichlet's theorem (or I guess I should call it the Prime Number Theorem for Arithmetic Progressions) for number fields. There I received a confirmation of the guess that primes are also "essentially equally distributed" among the residue classes of $K/qK$. On this note, Theorem 1 of the paper "On the Theorem of Barban and Davenport-Halberstam in Algebraic Number Fields" (Journal of Number Theory 13, 463-484, 1981) implies that for $\mathfrak{q}$ an integral domain of a number field $K$ and $\beta \in K$ an integer with $(\beta, \mathfrak{q}) = 1$, the number of primes in $K$ congruent to $\beta \pmod{\mathfrak{q}}$ with norm less than $X$ is</p> <p>$\frac{1}{\phi(\mathfrak{q})}I(1) + o(I(1)),$ </p> <p>where $I(1)$ is the analogue of the logarithmic integral. </p> <p>My next question: each prime in $K$ has a number of associates. Roughly, does each associate appear equally often?</p> <p>More precisely, my work is in $\mathbb{Z}[\omega]$ ($\omega$ a primitive cube root of unity) where there are $6$ associates of each prime. I can specify a specific associate by requiring it to be primary, i.e. $2 \pmod{3}$, and in the upper half plane. Now the above theorems imply that $\frac{1}{\phi (Nq)}(1+o(1))$ of the primes in $\mathbb{Z}[\omega]$ are congruent to a particular $\beta \pmod{q}$. Is it also true that one sixth of these are primary and in the upper half plane? Supposing that $3|q$ and $\beta \equiv 2 \pmod{3}$ so that such primes are automatically primary -- is it true that one half of primes in this congruence class are in the upper half plane? </p> http://mathoverflow.net/questions/29258/distribution-of-associate-primes-modulo-q-in-number-fields/29267#29267 Answer by Franz Lemmermeyer for Distribution of associate primes modulo q in number fields Franz Lemmermeyer 2010-06-23T18:41:05Z 2010-06-23T18:41:05Z <p>In the case you're interested in the answer is yes: just replace the modulus $q$ by $q\pi^3$, where $\pi$ generates the prime ideal with norm $3$. In general number fields, the question is not meaningful as it stands because the unit group has infinite order except when $K$ is complex quadratic. </p>