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. Chowla in the special case when the discriminant $d = p \equiv 5\,{\rm mod}\, 8$ is a prime. We use another form of class number formula and transform it using Dirichlet's $1/4$-th character sums. Our result elucidates other generalizations of the class number formula by Mitsuhiro, Nakahara and Uhera for general real quadratic fields.
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
\[
\frac{1}{k!}\sum_{n \leq x}\frac{n}{\phi(n)} \left( 1-\frac{n}{x} \right)^k
= M_k(x) + E_k(x).
\]
The upper bound for $\left | E_k(x) \right |$ established here thus improves the earlier known upper bound when $k=1$.