WebHere h (− k) denotes the class number (i.e., the number of equivalent classes of fractional ideals) of the imaginary quadratic field Q [√ − k]. Class numbers make an appearance in many of the theorems that will follow, and so one can think of them as representations of class numbers by trigonometric sums. WebWe prove several old and new theorems about finite sums involving characters and trigonometric functions. These sums can be traced back to theta function identities from Ramanujan's notebooks and were first systematically studied by Berndt and Zaharescu where their proofs involved complex contour integration.

arXiv:0804.0645v2 [math.NT] 4 Jun 2008

WebDec 1, 2024 · In this paper, without any appeal to theta functions, several classes of finite trigonometric sums, including the aforementioned sums, are evaluated in closed form … WebIf a sum cannot be carried out explicitly by adding up a finite number of terms, Sum will attempt to find a symbolic result. ... Some trigonometric sums with exponential … super idol lyrics to raise social credit

C. M. da Fonseca, Victor Kowalenko

WebFeb 18, 2024 · The mathematics literature contains many generalized trigonometric sums which are evaluated through contour integration methods, algebraic methods or through discrete Fourier analysis methods. The purpose of this paper is to show how Ramanujan’s theory of theta functions can be efficiently employed to evaluate certain generalized … WebJul 13, 2004 · Explicit evaluations of finite trigonometric sums arose in proving certain theta function identities of Ramanujan. In this paper, without any appeal to theta functions, several classes of finite trigonometric sums, including the aforementioned sums, are evaluated … WebAbstract. In this article we develop a general method by which one can explicitly evaluate certain sums of nth powers of products of d ≥ 1 elementary trigonometric functions evaluated at m = ( m 1 , … , m d )-th roots of unity. super idol 的笑容都没你的甜 lyrics