An Alternative Approach to Computing β(2k+1)
Date:
This was the first talk of the time I was at Bowdoin. The project was done during my sophomore summer under the supervision of Professor Naomi Tanabe. The poster presentation itself happened in October of my Junior year. However, at that time, we had not found the integral representation of the Dirichlet \(\beta\)-function evaluated at even integers. The work was finally completed and submitted in May 2021 and currently under review.
Abstract
In Ciaurri et al. work in 2015, the authors introduced a simple proof of Euler’s formula for \(\zeta(2k)\) and slightly modified the proof to derive an integral representation for \(\zeta(2k+1)\), where k is any positive integer. In this paper, by adapting their ideas, we present a new proof for evaluating the special values of the Dirichlet beta function, \(\beta(2k+1)\). Our approach relies on some properties of the Euler numbers and polynomials, and uses basic calculus and telescoping series. By a similar procedure, we also yield an integral representation of \(\beta(2k)\).
More Information
I have made this poster to accompany my poster presentation at the President’s Symposium in October 2019. The work is now published on Integers, feel free to check it out here.
Many Thanks to
Professor Naomi Tanabe who suggested me this topic and provided me a lot of knowledge in number theory I had never experienced before.
Here is the reference of the paper I mentioned in the abstract: O. Ciaurri, L. M. Navas, F. J. Ruiz, and J. L. Varona, A simple computation of \(\zeta(2k)\). Amer. Math. Monthly 122 (2015), no. 5, 444–451.