Formulas for Chebotarev densities of Galois extensions of number fields

摘要

We generalize the Chebotarev density formulas of Dawsey (Res Number Theory 3:27, 2017) and Alladi (J Number Theory 9:436–451, 1977) to the setting of arbitrary finite Galois extensions of number fields L ?/? K . In particular, if $$C subset G = {{mathrm{Gal}}}(L/K)$$ C ? G = Gal ( L / K ) is a conjugacy class, then we establish that the Chebotarev density is the following limit of partial sums of ideals of K : $$egin{aligned} -lim _{Xightarrow infty } sum _{egin{array}{c} 2le N(I)le X I in S(L/K; C) end{array}} rac{mu _K(I)}{N(I)} = rac{|C|}{|G|}, end{aligned}$$ - lim X → ∞ ∑ 2 ≤ N ( I ) ≤ X I ∈ S ( L / K ? C ) μ K ( I ) N ( I ) = | C | | G | , where $$mu _K(I)$$ μ K ( I ) denotes the generalized M?bius function and S ( L ?/? K ;? C ) is the set of ideals $$Isubset mathcal {O}_K$$ I ? O K such that I has a unique prime divisor $$mathfrak {p}$$ p of minimal norm and the Artin symbol $$left[ rac{L/K}{mathfrak {p}}ight] $$ L / K p is C . To obtain this formula, we generalize several results from classical analytic number theory, as well as Alladi’s concept of duality for minimal and maximal prime divisors, to the setting of ideals in number fields.