Elementary Trigonometric Sums related to Quadratic Residues

TL;DR

This paper explores number-theoretical properties of certain trigonometric sums related to quadratic residues for primes congruent to 3 mod 4, revealing connections with class numbers and providing explicit formulas.

Contribution

It introduces new relationships between trigonometric sums and quadratic residues, and establishes a simple formula linking these sums to class numbers h(-p).

Findings

T(p) equals p times the excess of odd over even quadratic residues.

C(p) relates to the class number h(-p).

Explicit formulas connect sums to quadratic residue properties.

Abstract

Let p be a prime = 3 (mod 4). A number of elegant number-theoretical properties of the sums T(p) = \sqrt{p}sum_{n=1}^{(p-1)/2} tan(n^2\pi/p) and C(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 in the set {1,2,...,p-1}; this number is positive if p = 3 (mod 8) and negative if p = 7 (mod 8). In this revised version the connection of these sums with the class-number h(-p) is also discussed. For example, a very simple formula expressing h(-p) by means of the aforementioned excess is proved. The bibliography has been considerably enriched. This article is of an expository nature.