Congruences for class numbers of $$\mathbb {Q}(\sqrt{\pm 2p})$$ when $$p\equiv 3$$ $$(\text {mod }4)$$ is prime

For a prime \(p\equiv 3\) \((\text {mod }4)\), let \(h(-8p)\) and h(8p) be the class numbers of \(\mathbb {Q}(\sqrt{-2p})\) and \(\mathbb {Q}(\sqrt{2p})\), respectively. Let \(\Psi (\xi )\) be the Hirzebruch sum of a quadratic irrational \(\xi \). We show that \(h(-8p)\equiv h(8p)\Big (\Psi (2\sqrt{2p})/3-\Psi (\frac{1+\sqrt{2p}}{2})/3\Big )\) \((\text {mod }16)\). Also, we show that \(h(-8p)\equiv 2\,h(8p)\Psi (2\sqrt{2p})/3\) \((\text {mod }8)\) if \(p\equiv 3\) \((\text {mod }8)\), and \(h(-8p)\equiv \big (2\,h(8p)\Psi (2\sqrt{2p})/3\big )+4\) \((\text {mod }8)\) if \(p\equiv 7\) \((\text {mod }8)\).