Functional distribution of $L(s, \chi _d)$ with real characters and denseness of quadratic class numbers
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- by Hidehiko Mishou and Hirofumi Nagoshi PDF
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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.References
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Additional Information
- Hidehiko Mishou
- Affiliation: Graduate School of Mathematics, Nagoya University, Chikusa-ku, Nagoya 464-8602, Japan
- Email: m98018a@math.nagoya-u.ac.jp
- Hirofumi Nagoshi
- Affiliation: Department of Mathematics, Faculty of Science, Niigata University, Niigata 950-2181, Japan
- Email: nagoshih@ybb.ne.jp
- 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.
- © Copyright 2006
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 358 (2006), 4343-4366
- MSC (2000): Primary 11M06, 41A30; Secondary 11M20, 11R29
- DOI: https://doi.org/10.1090/S0002-9947-06-03825-6
- MathSciNet review: 2231380