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Lower bounds for relative class numbers of imaginary abelian number fields and CM-fields. (English) Zbl 1122.11053
The author gives upper bounds for $$|L(1,\chi)|$$ where $$\chi$$ is a primitive Dirichlet character modulo $$f>1$$ and as applications bounds for relative class numbers $$h_K^-$$ of CM-fields $$K$$, especially in the case that $$K$$ contains an imaginary quadratic subfield. In order to describe one of the main results (Thm. 1) let $$\gamma$$ denote Euler’s constant and set $$\kappa_0:=2+\gamma-\log(4\pi)$$, $$\kappa_1:=2+\gamma-\log(\pi)$$. Then the following estimate is obtained: $\bigl |L(1,\chi\bigr| \leq\frac 12\bigl(\log(f)+\kappa_\chi\bigr),$ where $$\kappa_\chi:= \kappa_0$$, if $$\chi$$ is even, and $$\kappa_\chi:=\kappa_1$$, if $$\chi$$ is odd. Moreover, if $$\chi$$ is quadratic, then $$0<\beta<1$$ and $$L(\beta, \chi)=0$$ imply $0<L(1,\chi)\leq\frac{1-\beta}{8}\log^2(f).$ The author describes his method of proof as follows: “First, we start with the integral representations of $$L$$-functions $$L(s,\chi)$$ (used to prove their functional equations) and use inverse Mellin transforms to obtain bounds on $$|L(1,\chi)|$$ as integrals on the vertical line $$\operatorname{Re}(s)=c<1$$ of the complex plane of complex-valued functions (see Proposition 5). Second, we move these vertical lines of integration leftwards to $$\operatorname{Re}(s)=-\frac 12$$. In the process we pick up residues (see Lemma 7), which yields the map part of our upper bounds. Finally, to comiplete the proof of Theorem 1, we give an explicit bound on the modulus of these integrals on the line $$\operatorname{Re}(s)=-\frac 12$$ (see Lemma 8)”.

##### MSC:
 11M06 $$\zeta (s)$$ and $$L(s, \chi)$$ 11M20 Real zeros of $$L(s, \chi)$$; results on $$L(1, \chi)$$ 11R20 Other abelian and metabelian extensions 11R29 Class numbers, class groups, discriminants 11R42 Zeta functions and $$L$$-functions of number fields
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