Generalization of Bombieri's theorem and its applications
The authors generalize the Bombieri-Vinogradov mean value theorem to a class of arithmetic functions \(f\), which can roughly be described as follows. For \(n,q\leq x\), \(\chi\bmod q\), and \(\text{Re} (s)>1\) \[ \sum^\infty _{\substack{ n=1\\ (n,m)=1 }} f(n) \chi (n) n^{-s}= G_0 (m, s, \chi) L^{\alpha_1} (s, \chi) \cdot (L' (s, \chi) )^{\alpha_2}+ G_1 (m, s, \chi) \] (\(\alpha_1\in \mathbb{C}\), \(\alpha_2> 0\); \(G_0\), \(G_1\) regular for \(\sigma> {1\over 2}\), and satisfying some growth conditions). Then the mean value theorem holds for \(f\) with \(Q= x^{1/2- \varepsilon}\) \((\varepsilon> 0)\). The proof follows the well known analytical lines. In particular the theorem is applicable to the characteristic function of the set of integers \(n= a^2+ b^2\) \((a,b\in \mathbb{Z})\). (This has been shown by \textit{M. Gottschalk} and \textit{K.-H. Indlekofer} [J. Reine Angew. Math. 297, 65-79 (1978; Zbl 0363.10030)]\ with the slightly better \(Q= x^{1/2} (\ln x)^{-A}\).) As a second application the authors prove asymptotic formulae for sums for the type \[ \sum_{q\leq x} a(q) \sum_{\chi \bmod q} |L(1, \chi) |^{-2\alpha} \qquad (\alpha>0). \] Problems like this have been studied by \textit{W. Zhang} in various papers [Chin. Ann. Math., Ser. A 14, 1-5 (1993; Zbl 0779.11036)].
scientific article; zbMATH DE number 850855