In this note we give an alternate expression of class number formula for real quadratic fields with discriminant d≡5mod8. %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≡5mod8 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 ϕ(n) denote the Euler-totient function. We study the error term of the general k-th Riesz mean of the arithmetical function nϕ(n) for any positive integer k≥1, namely the error term Ek(x) where1k!∑n≤xnϕ(n)(1−nx)k=Mk(x)+Ek(x).
The upper bound for |Ek(x)| established here thus improves the earlier known upper bound when k=1.