On the Goldbach problem in algebraic number fields and the positivity of the singular integral (Q1364540)

The main purpose of this paper is to show the positivity of a generalized Dirichlet integral (a so-called singular integral) which arises in asymptotic formulae in connection with Waring's problem and the Goldbach-Vinogradov problem in the setting of an algebraic number field. Following the generalized Vinogradov-Vaughan method introduced by \textit{T. Mitsui} in [J. Math. Soc. Japan 12, 290-324, 325-372 (1960; Zbl 0152.03303)] the author gives an application to the Goldbach-Vinogradov theorem in number fields. He obtains in particular that the leading term in this formula has a positive coefficient.