Explicit formulas for Hecke Gauss sums in quadratic number fields
Let \(K=\mathbb{Q}(\sqrt{D})\), \(G(\omega)=\sum\limits_{\mu\bmod a}e^{2\pi i\,\text{Tr}(\mu^{2}\omega/\sqrt{D})}\). The following result is obtained: Let \(\omega\) be a nonzero element of \(K\), and let \(M\) be the smallest positive rational integer such that \(M\omega\) is integral. Then \[ G(\omega)/\sqrt{\mathbb{N}(a\;\gcd(2,a))}=\frac{1}{\sqrt{u}}\sum\limits_{\Delta_{1}}\left(\frac{\Delta_{1}}{A}\right) \left(\frac{4\mathbb{N}(\beta)/\Delta_{1}}{N/|\Delta_{1}|}\right)\sqrt{\text{sgn}(\Delta_{1})}. \] Some notations and implicit statements used in the theorem are explained in the paper. The proofs use only elementary algebraic manipulations.
scientific article; zbMATH DE number 5821525