Volume 10 - 1987

1. A remark on $\zeta(1+it).$

K Ramachandra.
Let $T\geq1000$ and $X = \exp(\log\log T/\log\log\log T)$. Consider any set $O$ of disjoint open intervals $I$ of length $1/X$, contained in the interval $T\leq t\leq T+e^X$. We prove in this paper, that $\vert\log\zeta(1+it)\vert\leq\varepsilon\log\log T$ in $O$ with the exception of $K$ intervals $I$, where $0<\varepsilon\leq1$ and $K$ depends only on $\varepsilon$.

2. Srinivasa Ramanujan (The inventor of the circle method) (22-12-1887 to 26-4-1920)

K Ramachandra.
Ramanujan's first letter to Hardy states an asymptotic formula for the coefficient of $x^n$ in the expansion of $(\sum_{m=-\infty}^{\infty}(-1)^mx^{m^2})^{-1}$ which can be regarded as the genesis of the circle method. In this paper, we try to give some indication of the possible intuition of Ramanujan in his discovery of the circle method. We discuss briefly the Goldbach, Waring and partition problems. At the end of the paper, there is a brief discussion on Ramanujan's $\tau$ function