Analytic Aspects of the Riemann Zeta and Multiple Zeta Values

Lupu, Cezar (2018) Analytic Aspects of the Riemann Zeta and Multiple Zeta Values. Doctoral Dissertation, University of Pittsburgh. (Unpublished)

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Abstract

This manuscript contains two parts. The first part contains fast converging series representations involving $\zeta(2n)$ for Apery's constant $\zeta(3)$. These representations are obtained via Clausen acceleration formulae. Moreover, we also find evaluations for more general rational zeta series involving $\zeta(2n)$ and binomial coefficients.

The second part will be devoted to the multiple zeta and special Hurwitz zeta values (multiple $t$-values). In this part, using a new approach involving integer powers of $\arcsin$ which come from particular values of the Gauss hypergeometric function, we are able to provide new proofs for the evaluations of $\zeta(2, 2, \ldots, 2)$, and $t(2, 2, \ldots, 2)$. Moreover, we are able to evaluate $\zeta(2, 2, \ldots, 2, 3)$, and $t(2, 2, \ldots, 2, 3)$ in terms of rational zeta series involving $\zeta(2n)$. On the other hand, using properties of the Clausen functions we can express these rational zeta series as a finite $\mathbb{Q}$-linear combinations of powers of $\pi$ and odd zeta values. In particular, we deduce the famous formula of Zagier for the Hoffman elements in a special case.

Zagier's formula is a remarkable example of both strength and the limits of the motivic formalism used by Brown in proving Hoffman's conjecture where the motivic argument does not give us a precise value for the special multiple zeta values $\zeta(2, 2, \ldots, 2, 3, 2, 2,\ldots, 2)$ as rational linear combinations of products $\zeta(m)\pi^{2n}$ with $m$ odd.
In \cite{Zagier1} the formula is proven indirectly by computing the generating functions of both sides in closed form and then showing that both are entire functions of exponential growth and that they agree at sufficiently many points to force their equality.

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Details

Item Type: University of Pittsburgh ETD
Status: Unpublished
Creators/Authors:
Lupu, Cezarcel47@pitt.educel47
ETD Committee:
Committee ChairHajlasz, Piotrhajlasz@pitt.eduhajlasz
Committee MemberTroy, William C.troy@math.pitt.edutroy
Committee MemberHales, Thomas C.hales@pitt.eduhales
Committee MemberMuscalu, Camilcamil@math.cornell.edu
Committee MemberSparling, Georgesparling@pitt.edusparling
Date: 21 October 2018
Date Type: Publication
Defense Date: 25 June 2018
Approval Date: 21 October 2018
Submission Date: 6 September 2018
Access Restriction: No restriction; Release the ETD for access worldwide immediately.
Number of Pages: 86
Institution: University of Pittsburgh
Schools and Programs: Dietrich School of Arts and Sciences > Mathematics
Degree: PhD - Doctor of Philosophy
Thesis Type: Doctoral Dissertation
Refereed: Yes
Uncontrolled Keywords: Riemann zeta function, rational zeta series, multiple zeta values, multiple special Hurwitz zeta values, Gauss hypergeometric function, Apery's constant, Zagier's formula for Hoffman elements
Date Deposited: 22 Oct 2018 01:04
URI: http://d-scholarship.pitt.edu/id/eprint/35330

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