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On Siegel zeros of Hecke-Landau zeta-functions. (English) Zbl 0838.11075

Let \(K\) be an algebraic number field and let \(\zeta_K (s, \chi)\) denote the Hecke zeta function associated with a Hecke character \(\chi \pmod {\mathfrak q}\). Denote by \(\chi_1\) the possible exceptional real character \(\pmod {\mathfrak q}\), if it exists. In the first part of the paper the authors study relations between the lower estimates of \(\zeta_K (1, \chi_1)\) and the location of the real zeros of \(\zeta_K (s, \chi_1)\). The results are analogous to those known for Dirichlet \(L\)-functions. In the second part of the paper the authors establish a Brun-Titchmarsh type estimate for the number of prime ideals in a fixed narrow ideal class. Finally they observe that the reduction of the factor 2 in this inequality would have important arithmetic consequences. So again the situation in number fields is very similar to what is known in the classical case of arithmetic progressions.

MSC:

11R42 Zeta functions and \(L\)-functions of number fields
11M41 Other Dirichlet series and zeta functions
11N36 Applications of sieve methods
11R45 Density theorems
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