On the divisor function and class numbers of real quadratic fields. II (Q757472)

This paper is a continuation of the author's preceding one [Proc. Japan Acad., Ser. A 66, No. 5, 109--111 (1990; Zbl 0714.11068)] in which he provided lower bounds for the class number of real quadratic fields of narrow Richaud-Degert type in terms of the divisor function \(\tau(x)\). In this paper, he proves the following for the class number \(h(d)\) of real quadratic fields \(\mathbb Q(\sqrt{d})\) \((d=\ell^ 2+r)\) of E.R.D. type: Let \[ P=\{\text{primes }p_ i:\;p_ i\mid \ell \text{ with } (r,p_ i)=1,\quad r\not\equiv 1\pmod{p_ i}\} \] and put \[ P(\Delta)=\left\{s=\prod_{i}p_ i^{e_ i}: e_ i\geq 1, s\leq \sqrt{\Delta /2}\right\} \] (\(\Delta\) : discriminant of \(\mathbb Q(\sqrt{d}))\). Then \(h(d)\geq \tau(q)\) for all \(q\in P(\Delta)\).