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Transactions of the American Mathematical Society

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Functional distribution of $ L(s, \chi_d)$ with real characters and denseness of quadratic class numbers

Authors: Hidehiko Mishou and Hirofumi Nagoshi
Journal: Trans. Amer. Math. Soc. 358 (2006), 4343-4366
MSC (2000): Primary 11M06, 41A30; Secondary 11M20, 11R29
Published electronically: May 17, 2006
MathSciNet review: 2231380
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Abstract: We investigate the functional distribution of $ L$-functions $ L(s, \chi_d)$ with real primitive characters $ \chi_d$ on the region $ 1/2 < \operatorname{Re} s <1$ as $ d$ varies over fundamental discriminants. Actually we establish the so-called universality theorem for $ L(s, \chi_d)$ in the $ d$-aspect. From this theorem we can, of course, deduce some results concerning the value distribution and the non-vanishing. As another corollary, it follows that for any fixed $ a, b$ with $ 1/2< a< b<1$ and positive integers $ r', m$, there exist infinitely many $ d$ such that for every $ r=1, 2, \cdots, r'$ the $ r$-th derivative $ L^{(r)} (s, \chi_d)$ has at least $ m$ zeros on the interval $ [a, b]$ in the real axis. We also study the value distribution of $ L(s, \chi_d)$ for fixed $ s$ with $ \operatorname{Re} s =1$ and variable $ d$, and obtain the denseness result concerning class numbers of quadratic fields.

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Additional Information

Hidehiko Mishou
Affiliation: Graduate School of Mathematics, Nagoya University, Chikusa-ku, Nagoya 464-8602, Japan

Hirofumi Nagoshi
Affiliation: Department of Mathematics, Faculty of Science, Niigata University, Niigata 950-2181, Japan

Received by editor(s): January 3, 2004
Received by editor(s) in revised form: August 9, 2004
Published electronically: May 17, 2006
Additional Notes: Both authors were supported by the JSPS Research Fellowships for Young Scientists.
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.