On the prehistory of growth of groups

The subject of growth of groups has been active in the former Soviet Union since the early 50's and in the West since 1968, when articles of \v{S}varc and Milnor have been published, independently. The purpose of this note is to quote a few articles showing that, before 1968 and at least retrospectively, growth has already played some role in various subjects.Comment: 12 page

The Circle Problem of Gauss and the Divisor Problem of Dirichlet-Still Unsolved

Let r(2)(n) denote the number of representations of the positive integer n as a sum of two squares, and let d(n) denote the number of positive divisors of n. Gauss and Dirichlet were evidently the first mathematicians to derive asymptotic formulas for Sigma(n <= x) r(2)(n) and Sigma(n <= x) d(n), respectively, as x tends to infinity. But what is the error made in such approximations? Number theorists have been attempting to answer these two questions for over one and one-half centuries, and although we think that we essentially know what these errors are, progress in proving these conjectures has been agonizingly slow. Ramanujan had a keen interest in these problems, and although, to the best of our knowledge, he did not establish any bounds for the error terms, he did give us identities that have been used to derive bounds, and two further identities that might be useful, if we can figure out how to use them. In this paper, we survey what is known about these two famous unsolved problems, with a moderate emphasis on Ramanujan's contributions