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.


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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