The Hooley–Huxley Contour Method for Problems in Number Fields

The Hooley-Huxley contour method for problems in number fields III : frobenian functions

“…This differs from the definition in [1] and [2] in that we have not excluded ideals with prime divisors that ramify in L.…”

“…In the earlier papers of this series, [1] and [2], we applied the Hooley-Huxley contour method, as described in [9], to sums of arithmetic functions defined on the integral ideals of a number field K, say. The contour method allows the restriction of the sums to small regions of ideals, S (x, ψ 0 , ) , defined below.…”

“…In the present paper we introduce the ideas of Odoni, see [7] for instance, and, assuming the arithmetic functions are finite-valued, examine when these functions take a given value. It will ease reading of this paper to have [1] and [2] to hand.…”

“…Proof of Theorem 1 From sections 4 and 5 of [1] and the references therein it can be seen that Θ(S, γ) differs from…”

“…The product of L-functions occurring in (12) is of exactly the form to which we can apply the Hooley-Huxley method though none of our previous applications have all the features of (12). In [1] the Hooley-Huxley method is applied to integrals containing products as in (12) though the L-functions occurring have only E = K. In [2] we have L-functions of the type (9) but with no logarithms of Lfunctions. Nonetheless the methods of [2] give the following version of part of Theorem 1 of [1].…”