A determinant involving Ramanujan sums and So's conjecture (Q1984243)

Let \(m\), \(n\) integers and \(c(m,n)= \sum\limits^n_{\substack{h=1\\ (h,n)=1}} e^{2\pi i\frac{m}{n}h}\) the Ramanujan sum and \(D(n)\) the set of divisors of \(n\). The author computes the determinant of the matrix \(C_n\) with the elements \(c(d,t)\), \(d,t\in D(n)\). The value is the astonishing simple \(\pm n^{\frac{1}{2}\tau(n)}\), where \(\tau(n)\) is the divisor function. He uses this computation to prove a weak version of So's conjecture on circulant graphs with integral spectrum.