# Rational zeta series for $\zeta(2n)$ and $\zeta(2n+1)$

Orr, Derek (2019) Rational zeta series for $\zeta(2n)$ and $\zeta(2n+1)$. Master's Thesis, University of Pittsburgh. (Unpublished)

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## Abstract

I will begin by using the cotangent function to find rational zeta series with $\zeta(2n)$ in terms of $\zeta(2k+1)$ and $\beta(2k)$, the Dirichlet beta function. I then develop a certain family of generalized rational zeta series using the generalized Clausen function $\Clausen_{m}(x)$ and use those results to discover a second family of generalized rational zeta series. As a special case of my results from Theorem 3.1, I prove a conjecture given in 2012 by F.M.S. Lima. Later, I use the same analysis but for the digamma function $\psi(x)$ and negapolygammas $\psi^{(-m)}(x)$. With these, I extract the same two families of generalized rational zeta series with $\zeta(2n+1)$ on the numerator rather than $\zeta(2n)$. Afterwards, I look into the applications of these rational zeta series and how they are related to other special functions such as the multiple zeta function.

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## Details

Item Type: University of Pittsburgh ETD
Status: Unpublished
Creators/Authors:
Orr, Derekdjo15@pitt.edudjo15
ETD Committee:
Committee ChairSparling, Georgegnilraps@gmail.com
Committee MemberHales, Thomashales@pitt.eduhales
Committee MemberHajlasz, Piotrhajlasz@pitt.eduhajlasz
Date: 19 June 2019
Date Type: Publication
Defense Date: 25 October 2018
Approval Date: 19 June 2019
Submission Date: 18 December 2018
Access Restriction: No restriction; Release the ETD for access worldwide immediately.
Number of Pages: 41
Institution: University of Pittsburgh
Schools and Programs: Dietrich School of Arts and Sciences > Mathematics
Degree: MS - Master of Science
Thesis Type: Master's Thesis
Refereed: Yes
Uncontrolled Keywords: Riemann zeta function, Dirichlet beta function, Clausen integral, negapolygammas, rational zeta series, polygamma function
Date Deposited: 19 Jun 2019 20:33
URI: http://d-scholarship.pitt.edu/id/eprint/35884

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• Rational zeta series for $\zeta(2n)$ and $\zeta(2n+1)$. (deposited 19 Jun 2019 20:33) [Currently Displayed]