On the order of Hecke-Landau zeta functions near the line \(\sigma =1\)
Let K be a fixed algebraic number field of degree n over the rationals with discriminant d. There are problems in the theory of algebraic numbers for which we need some information about the growth of the Hecke- Landau zeta-function \(\zeta_ K(\sigma +it,\chi)\), \(\chi\) mod \({\mathfrak q}\), near the line \(\sigma =1\) as \(t\to \infty\). In the present paper the authors derive for the region \(1-(n+1)^{-1}\leq \sigma \leq 1\), \(t\geq 1.1\) the estimate \[ | \zeta_ K(\sigma +it,\chi)| \leq c_ 1\cdot N{\mathfrak q}^{c_ 2}\cdot t^{c_ 3(1-\sigma)^{3/2}}\cdot (\log t)^{2/3}\quad, \] where \(c_ 1\), \(c_ 2\) and \(c_ 3\) can be expressed in terms of n and d with pure numerical constants. Specially, we have \(c_ 2=1.03\cdot 10^ 5\cdot n^ 5\), which seems to be far from best possible. The proof is based on some estimates connected with the extension of I. M. Vinogradov's method to the theory of Hecke-Landau zeta-functions.
scientific article; zbMATH DE number 3991627