A proof of the Landsberg-Schaar relation by finite methods (Q829850)

The article gives a proof of the Landsberg-Schaar relation: for positive integral \(a\) and \(b\) \[ \frac{1}{\sqrt{a}} \sum_{n=0}^{a-1} \exp \left(\frac{2 \pi i n^{2} b}{a}\right)=\frac{1}{\sqrt{2 b}} \exp \left(\frac{\pi i}{4}\right) \sum_{n=0}^{2 b-1} \exp \left(-\frac{\pi i n^{2} a}{2 b}\right). \] The proof is based only on techniques of elementary number theory (Gauss sums, Hensel lemma, finite Fourier series).