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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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