Methods to Find the Closed form of ζ(2) and ζ(2n)

Abstract

Finding the closed form of the celebrated special function is not an easy task. For this purpose, this article deals with the Riemann zeta function. The Riemann zeta function is one of the most important functions in mathematics, and it mainly relates to the area of analytic number theory. Based on this function, the Riemann hypothesis was also proposed. This function is defined as , with . In this article, several methods are provided to calculate specific values of Riemann zeta function. Specifically, the value when and, more generally, when . Finding the value of the function at two was used to be a world-class challenge, named Basel Problem. No one had solved this problem until Euler appeared, and since then many ways had appeared. In this article, two ways to solve the Basel problem are firstly provided. Afterwards, method involving Bernoulli numbers to calculate all the -values at even arguments is also shown.

Locations

• Highlights in Science Engineering and Technology - View - PDF