# Some arithmetic problems related to class group \(L\)–functions

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## Abstract

We prove that for each fundamental discriminant \(-D<0\), there exists at least one ideal class group character \(\chi \) of \(\mathbb {Q}(\sqrt{-D})\) such that the \(L\)–function \(L(\chi ,s)\) is nonvanishing at \(s=\tfrac{1}{2}\). In addition, assuming that the quadratic Dirichlet \(L\)-function \(L(\chi _D,\tfrac{1}{2}) \ge 0\), we prove that the class number \(h(-D)\) satisfies the effective lower bound for each fundamental discriminant \(-D < 0\) with \(D \ge (8 \pi /e^{\gamma })^{(\frac{1}{2}-\varepsilon )^{-1}}\) where \(0 < \varepsilon < 1/2\) is arbitrary and fixed (here \(\gamma \) is Euler’s constant).

$$\begin{aligned} h(-D) \ge 0.1265 \cdot \varepsilon D^{\frac{1}{4}}\log (D) \end{aligned}$$

## Keywords

\(L\)-functions Ideal class group Class number problem## Mathematics Subject Classification

11M41 11F67## Notes

### Acknowledgments

This work was completed during the 2013 REU in Number Theory at Texas A&M University. The authors thank the NSF for their generous support. The authors also thank Riad Masri for persistent encouragement and advice. The authors thank the referee for their careful reading of the manuscript and many helpful suggestions.

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