Hirzebruch sum and class number of the quadratic fields (Q1179949)

For a real quadratic irrational number \(\beta\), which has a simple continued fraction expansion \([\hat a_ 0,\dots,\hat a_ s,\overline{a_ 1,\dots,a_ k}]\) with basic period \(\overline{a_ 1,\dots,a_ k}\), the author defines the Hirzebruch sum \(\psi(\beta)=\sum_{j=1}^ k (-1)^{j+s} a_ j\) if \(k\) is even and \(\psi(\beta)=0\) if \(k\) is odd. Let \(h(x)\) denote the class number of the quadratic field \(\mathbb{Q}(\sqrt{x})\). Suppose both \(d>1\) and \(-k<-1\) be the fundamental discriminants with \((d,k)=1\). An expression of \(h(-k)h(-dk)\) by means of Hirzebruch sums is proved in the paper for the cases: \(2\nmid k\), \(4\| k\) and \(8\| k\). As an example of the results, the author shows that if \(p\equiv 1\pmod 8\) is a prime with \(h(p)=1\), then \(3h(- 4p)=\psi(\sqrt{4p})\).