In this note we give an alternate expression of class number formula for real quadratic fields with discriminant $d \equiv 5\, {\rm mod}\, 8$.
%Dirichlet's classical class number formula for real quadratic fields expresses `class number' in somewhat `transcend' manner, which was simplified by P. […]

Let $\phi(n)$ denote the Euler-totient function. We study the error term of
the general $k$-th Riesz mean of the arithmetical function
$\frac {n}{\phi(n)}$ for any positive integer $k \ge 1$, namely the error
term $E_k(x)$ where
\[
\frac{1}{k!}\sum_{n \leq x}\frac{n}{\phi(n)} \left( […]