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Upper bounds on residues of Dedekind zeta functions of non-normal totally real cubic fields. (English) Zbl 1470.11297

It has been shown by the author (Theorem 8 in [Manuscr. Math. 125, No. 1, 43–67 (2008; Zbl 1137.11072)]) that if \(K\) is a non-normal totally real cubic field, then \[ \mathrm{Res}_{s=1}\zeta_K(s)\le c(\log d_K+5)^2, \] where \(c\le1\) depends explicitly on the splitting behaviour of \(2\) in \(K\) and \(d_K\) denotes the absolute value of the discriminant of \(K\).
Now he shows in Theorem 1.7 that if \(L\) the real quadratic subfield of the normal closure of \(K\) and \(d_K\gg d_L\log^2 d_L\), then for every prime \(p\) one has \[ \mathrm{Res}_{s=1}\zeta_K(s)\le \frac{c_p+o(1)}8\log\frac{d_K^2}{d_L}, \] for \(d_L\) tending to infinity, with \[ c_p=\left(1-1/p\right)^3\prod_{P|p\ \text{ in }\ K}\left(1-1/N(P)\right)^{-1}. \] This is achieved by an improvement of the effective evaluation of \(|L(1,\chi)|\) for primitive even Dirichlet characters.

MSC:

11R42 Zeta functions and \(L\)-functions of number fields
11M20 Real zeros of \(L(s, \chi)\); results on \(L(1, \chi)\)
11R11 Quadratic extensions
11R16 Cubic and quartic extensions

Citations:

Zbl 1137.11072
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References:

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