196 research outputs found
Elementary Trigonometric Sums related to Quadratic Residues
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.Comment: 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 Mat
Brauer characters and the Harris–Knörr correspondence in p-solvable groups
AbstractIf b is a p-block of a normal subgroup N of a p-solvable group G and b* is its Brauer correspondent in NN(L), where L is a defect group of b, then Harris–Knörr correspondents over b and b* contain equal numbers of irreducible Brauer characters over height zero Brauer characters
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