On Class Numbers of Real Quadratic Fields with Certain Fundamental Discriminants

Let N denote the sets of positive integers and D ∈ N be square free, and let χD, h = h(D) denote the non-trivial Dirichlet character, the class number of the real quadratic field K = Q(pD), respectively. Ono proved the theorem in [2] by applying Sturm’s Theorem on the congruence of modular form to Cohen’s half integral weight modular forms. Later, Dongho Byeon proved a theorem and corollary in [1] by refining Ono’s methods. In this paper, we will give a theorem for certain real quadratic fields by considering above mentioned studies. To do this, we shall obtain an upper bound different from current bounds for L(1,χD) and use Dirichlet’s class number formula.