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 (Unpublished)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Status:	Unpublished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:	Creators Email Pitt Username ORCID Orr, Derek djo15@pitt.edu djo15
ETD Committee:	Title Member Email Address Pitt Username ORCID Committee Chair Sparling, George gnilraps@gmail.com Committee Member Hales, Thomas hales@pitt.edu hales Committee Member Hajlasz, Piotr hajlasz@pitt.edu hajlasz
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
Last Modified:	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]

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