Über einige Formeln in analytischer Zahlentheorie. I. (On some formulae in analytic number theory. I) (Q1825244)

Let K be a number field and let us denote by \(\chi\) a Hecke character of finite order and conductor \({\mathfrak f}\). The author considers analytic properties of the function defined for \(Im(w)>0\) by the formula \(k(w,\chi)=\sum_{Im(w)>0}\exp (w\rho)\), the summation being taken over all non-trivial zeros of the Hecke zeta function \(\zeta_ K(s,\chi)\) lying on the upper half-plane. It is proved that k has analytic continuation to a meromorphic function on M, the Riemannian surface of log(z). Its only singularities are simple poles at the points \(w=(m\cdot \log (N_{K/{\mathbb{Q}}}({\mathfrak p})))\exp (2\pi i\ell)\) (m\(\neq 0\), \({\mathfrak p}^ a \)prime ideal, \({\mathfrak p} \nmid {\mathfrak f})\) and at some points of the form \(w=m\pi i \exp (\pi i\ell)\), \(m\neq 0\) (\(\ell\) denotes the number of the sheet of M containing w). Moreover k satisfies two functional equations on M. This is another instance of the mutual correlation between zeta-zeros and prime ideals usually observed in connection with so-called explicit formulae. In fact the author's results can be considered as a ``complex form'' of the Riemann-von Mangoldt formula. Details, however, are too involved to be reproduced here. The method of the proof closely follows the reviewer's paper [``The k-functions in multiplicative number theory, I'' (to appear in Acta Arith.)].