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.

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.

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