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

Authors

  • Wenfei Dong

DOI:

https://doi.org/10.54097/v5z00292

Keywords:

Riemann zeta function; Basel problem; Bernoulli numbers; Closed form.

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.

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References

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Published

29-03-2024

How to Cite

Dong, W. (2024). Methods to Find the Closed form of ζ(2) and ζ(2n). Highlights in Science, Engineering and Technology, 88, 504-508. https://doi.org/10.54097/v5z00292