The Hooley–Huxley contour method, for problems in number fields II: factorization and divisibility

Extract

Let L be a Galois extension of the number field K. Set n = n_K = deg K/ℚ, n_L = deg L/ℚ and n_L/K = deg L/K. Let I = I_L/K denote the group of fractional ideals of K whose prime decomposition contains no prime ideals that ramify in L, and let P = {(α)ΣI: αΣK*, α>0}. Following Hecke [9}, let (λ₁, λ₂, …, λ_{n − 1}) be a basis for the torsion-free characters on P that satisfy λ_i(α) = 1 (1≤i≤n − 1) for all units α>0 in , the ring of integers of K. Fixing an extension of each λ_i to a character on I, then λ_i,(α) (1 ≤i≤n − 1) are defined for all ideals α of K that do not ramify in L. So, for such ideals, we can define . Then the small region of K referred to above is

for 0<l<½ and with the notation that, for any α ∈ ℝ, we set β. where β is the unique real satisyfing .