K Chakraborty ; S Kanemitsu ; T Kuzumaki - On the class number formula of certain real quadratic fields

hrj:178 - Hardy-Ramanujan Journal, January 1, 2013, Volume 36 - https://doi.org/10.46298/hrj.2013.178
On the class number formula of certain real quadratic fields

Authors: K Chakraborty ; S Kanemitsu ; T Kuzumaki

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.


Volume: Volume 36
Published on: January 1, 2013
Submitted on: March 3, 2015
Keywords: Real quadratic field, class number formula,[MATH] Mathematics [math]


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