Some formulas related to Gauss's sum (Q2536839)

\textit{S. Chowla} [Tôhoku Math. J. 32, 352--353 (1929; JFM 56.0177.03)] has proved the formula \[ \sum_{s=0}^{n-1} (-1)^s e^{\pi in(2s+1)^2/(4m)} = \left(e^{\pi i/4}/\sqrt{mn}\right) \sum_{s=1}^{mn} e^{-\pi is^2/mn} \sec(\pi s/m), \tag{*} \] where \(m, n\) are arbitrary odd positive integers. The present paper contains an elementary proof of (*) and indeed of the slightly more general formula \[ e^{-\pi ia/m} \sum_{s=1}^{mn} e^{\pi ias^2/mn} = \left(\frac{-1}{mn}\right)^{(a' - 1)/2} \left(\frac{a}{mn}\right) \sqrt{mn} \sum_{k=0}^{n-1} (-1)^k \exp\{-\pi i(2k+1)^2 a'n/(4m)\} \] where \(m, n\) are odd, \((a,2mn) =1\), \(aa' \equiv 1\pmod{2mn}\) and \((a/mn)\) is the Jacobi symbol. In addition it is shown that \[ \sum_{s=1}^{mn} e^{2\pi ias^2/mn} \sec(2\pi s/m) = (-a/mn) \sqrt{mn} \sum_{k=-(n-1)/2}^{(n-1)/2} (-1)^k e^{2\pi ik^2a'n/m}. \]