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Spacing of zeros of Hecke $$L$$-functions and the class number problem. (English) Zbl 1007.11051
Assuming some hypothesis on a vertical distribution of zeros of a Hecke $$L$$-function, the authors obtain sharp effective bounds for the class number of an imaginary quadratic field. More precisely, let $$-q$$ be a fundamental discriminant, $$q>4$$. Let $$Cl(K)$$ be the group of ideal classes of $$K=\mathbb Q(\sqrt{-q})$$. Let $$L(s,\psi)=\sum_I\psi(I)(N I)^{-s}$$ for a class group character $$\psi\in \widehat{Cl}(K)$$, where $$I$$ runs over the non-zero integral ideals. Let $$\rho=1/2+i\gamma$$ denote the zeros of $$L(s,\psi)$$ on the critical line and $$\rho'=1/2+i\gamma'$$ denote the nearest zero to $$\rho$$ on the critical line ($$\rho=\rho'$$ if $$\rho$$ is a multiple zero). Let $$L(s, \chi)=\sum_{n=1}^\infty\chi(n)n^{-s}$$, where $$\chi(n)=(-q/n)$$ is the Kronecker symbol. Let $$A\geq 0$$ and $$\log T\geq (\log q)^{A+6}$$. The authors prove that $L(1,\chi)\geq(\log T)^{-2}(\log q)^{2A-6}$ provided $\#\left\{\rho; 2\leq\gamma\leq T, |\gamma-\gamma'|\leq{\pi(1-\alpha)\over\log\gamma} \right\}\geq \frac{cT\log T }{\alpha(\log q)^A}$ for some $$0<\alpha\leq 1$$, where $$c$$ is a large absolute constant. Also they obtain the similar result in the case $$\rho$$ and $$\rho'$$ are zeros of the Riemann zeta-function. Recently P. Sarnak and A. Zaharescu [Duke Math. J. 111, 495-507 (2002; Zbl 1008.11033)] investigated the lower bounds for $$L(1,\chi)$$ assuming some hypothesis on a horizontal distribution of zeros of appropriate $$L$$-functions. The lower bounds for the class number $$h=|Cl(K)|$$ follow by the Dirichlet formula $$h=\pi^{-1}\sqrt{q}L(1,\chi)$$.

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