# Volume 12 - 1989

### 1. A Lemma in complex function theory I

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 $\vert z\vert\leq r$ where $\vert f(z)\vert\leq M$ holds, and $A\geq1$, then $\vert f(0)\vert\leq(24A\log M) (\frac{1}{2r}\int_{-r}^r \vert f(iy)\vert\,dy)+M^{-A}.$ 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

In this sequel to the previous paper with the same title, we prove a similar result as in part I, but which holds for $\vert f(z)\vert^k$, where $k>0$ is any real number.

### 3. A trivial remark on Goldbach conjecture

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 $\Omega$-result related to $r_4(n)$.

Let $r_4(n)$ be the number of ways of writing $n$ as the sum of four squares. Set $P_4(x)= \sum \limits_{n\le x} r_4(n)-\frac {1}{2}\pi^2 x^2$, the error term for the average order of this arithmetical function. In this paper, following the ideas of ErdÃ¶s and Shapiro, a new elementary method is developed which yields the slightly stronger result $P_4(x)= \Omega_{+}(x \log \log x)$. We also apply our method to give an upper bound for a quantity involving the Euler $\varphi$-function. This second result gives an elementary proof of a theorem of H. L. Montgomery