Arithmetic Consequences of Jacobi's Two-Squares Theorem

Abstract

There is a well-known formula due to Jacobi for the number r₂(n) of representations of the number n as the sum of two squares. This formula implies that the numbers r₂(n) satisfy elegant arithmetic relations. Conversely, these arithmetic properties essentially imply Jacobi's formula. So it is of interest to give direct proofs of these arithmetic relations, and this we do.