Expressing a number as the sum of two coprime squares (Q1127847)

Let \(m\), \(n\) be positive integers. Let \(\psi(m) = m \prod_{p| m} (1 + (1/p))\) where the product is over all prime factors \(p\) of \(m\). Let \(c_{m,n}\) denote the number of pairs of coprime positive integers \((c,d)\) such that \(c^2 + d^2 \leq n\) and \(c\) is divisible by \(m\). The authors use two-dimensional hyperbolic geometry to show that the limit of \(c_{m,n}/n\) as \(n\) tends to infinity (exists and) is \(3/(2\pi \psi(m))\). They include another proof of the same fact, provided by the referee, which uses Eisenstein series and Ikehara's Tauberian Theorem.