Volume 12 - 1989


1. A Lemma in complex function theory I

R Balasubramanian ; K Ramachandra.
Continuing our earlier work on the same topic published in the same journal last year we prove the following result in this paper: If f(z) is analytic in the closed disc |z|r where |f(z)|M holds, and A1, then |f(0)|(24AlogM)(12rrr|f(iy)|dy)+MA. Proof uses an averaging technique involving the use of the exponential function and has many applications to Dirichlet series and the Riemann zeta function.

2. A Lemma in complex function theory II

R Balasubramanian ; K Ramachandra.
In this sequel to the previous paper with the same title, we prove a similar result as in part I, but which holds for |f(z)|k, where k>0 is any real number.

3. A trivial remark on Goldbach conjecture

K Ramachandra.
Using a variation on the circle method, we provide another proof of a theorem of Srinivasan in his paper titled ``A remark on Goldbach's problem''.

4. An Ω-result related to r4(n).

Sukumar Das Adhikari ; R Balasubramanian ; A Sankaranarayanan.
Let r4(n) be the number of ways of writing n as the sum of four squares. Set P4(x)=nxr4(n)12π2x2, the error term for the average order of this arithmetical function. In this paper, following the ideas of Erdös and Shapiro, a new elementary method is developed which yields the slightly stronger result P4(x)=Ω+(xloglogx). We also apply our method to give an upper bound for a quantity involving the Euler φ-function. This second result gives an elementary proof of a theorem of H. L. Montgomery