An analogue of Ramanujan’s formula for ζ(2m +1)

Abstract

Understanding the nature of the particular values of the infinite series ∞ n=1 1 ns has always been a difficult problem in number theory. Euler gave an elegant formula for even zeta values which implies that even zeta values are not only irrational but also transcendental. Nevertheless, the nature of odd zeta values remains open. An interesting formula for odd zeta values can be found in page 319 of the lost notebook of Ramanujan. In the same page, Ramanujan mentioned that the formula for odd zeta values can be obtained from one of his partial fraction decompositions of cotangent function and cotangent hyperbolic function. In this thesis, we study a few partial fraction decompositions for trigonometric functions given by Ramanujan. Utilizing these partial fraction decompositions and Cauchy integration technique, we establish Dirichlet character analogue of the formula for odd zeta values.