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

We investigate the functional distribution of L L -functions L ( s , χ d ) L(s, \chi _d) with real primitive characters χ d \chi _d on the region 1 / 2 > Re ⁡ s > 1 1/2 > \operatorname {Re} s >1 as d d varies over fundamental discriminants. Actually we establish the so-called universality theorem for L ( s , χ d ) L(s, \chi _d) in the d 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 a, b with 1 / 2 > a > b > 1 1/2> a> b>1 and positive integers r ′ , m r’, m , there exist infinitely many d d such that for every r = 1 , 2 , ⋯ , r ′ r=1, 2, \cdots , r’ the r r -th derivative L ( r ) ( s , χ d ) L^{(r)} (s, \chi _d) has at least m m zeros on the interval [ a , b ] [a, b] in the real axis. We also study the value distribution of L ( s , χ d ) L(s, \chi _d) for fixed s s with Re ⁡ s = 1 \operatorname {Re} s =1 and variable d d , and obtain the denseness result concerning class numbers of quadratic fields.