Functional distribution of 𝐿(𝑠,𝜒_{𝑑}) with real characters and denseness of quadratic class numbers

Mishou, Hidehiko; Nagoshi, Hirofumi

doi:10.1090/S0002-9947-06-03825-6

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

Trans. Amer. Math. Soc. 358 (2006), 4343-4366 Request permission

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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