Volume 36 - 2013


1. On the class number formula of certain real quadratic fields

K Chakraborty ; S Kanemitsu ; T Kuzumaki.
In this note we give an alternate expression of class number formula for real quadratic fields with discriminant d5mod8. %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=p5mod8 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.

2. On the Riesz means of nϕ(n)

A Sankaranarayanan ; Saurabh Kumar Singh.
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 k1, namely the error term Ek(x) where1k!nxnϕ(n)(1nx)k=Mk(x)+Ek(x).
The upper bound for |Ek(x)| established here thus improves the earlier known upper bound when k=1.

3. Ramanujan series for arithmetical functions

M. Ram Murty.
We give a short survey of old and new results in the theory of Ramanujan expansions for arithmetical functions.

4. I am fifty-five years old

K Ramachandra.
This is an autobiographical article published posthumously.