Elementary Trigonometric Sums related to Quadratic Residues

Let p be a prime = 3 (mod 4). A number of elegant number-theoreticalproperties of the sums T(p) = \sqrt{p}sum_{n=1}^{(p-1)/2} tan(n^2\pi/p) andC(p) = \sqrt{p}sum_{n=1}^{(p-1)/2} cot(n^2\pi/p) are proved. For example, T(p)equals p times the excess of the odd quadratic residues over the even ones inthe set {1,2,...,p-1}; this number is positive if p = 3 (mod 8) and negative ifp = 7 (mod 8). In this revised version the connection of these sums with theclass-number h(-p) is also discussed. For example, a very simple formulaexpressing h(-p) by means of the aforementioned excess is proved. Thebibliography has been considerably enriched.