2010/01/15 by A. Laradji, Laradji, A., M. Mignotte +5
Mathematics · #11A15 #11L03 #11L05 #Analytic Number Theory Research #FOS: Mathematics #History and Theory of Mathematics #Mathematics and Applications #Number Theory (math.NT) #math.NT #msc:11A15 #msc:11L03 #msc:11L05
paper · pdf · doi:10.48550/arxiv.1001.2638
A number of misprints have been corrected and one or two improvements have been done to the previous version of the paper with same title. The paper will appear to Elem. der Math
openalex publication_date 2010/01/15 · arxiv created 2012/05/18 · arxiv updated 2012/05/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let p be a prime = 3 (mod 4). A number of elegant number-theoretical properties of the sums T(p) = √(p)sumn=1(p-1)/2 tan(n2π/p) and C(p) = √(p)sumn=1(p-1)/2 cot(n2π/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.