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1. Kopp, Gene. Indefinite Theta Functions and Zeta Functions.

Degree: PhD, Mathematics, 2017, University of Michigan

URL: http://hdl.handle.net/2027.42/140957

We define an indefinite theta function in dimension g and index 1 whose modular parameter transforms by a symplectic group, generalizing a construction of Sander Zwegers used in the theory of mock modular forms. We introduce the indefinite zeta function, defined from the indefinite theta function using a Mellin transform, and prove its analytic continuation and functional equation. We express certain zeta functions attached to ray ideal classes of real quadratic fields as indefinite zeta functions (up to gamma factors). A Kronecker limit formula for the indefinite zeta function – and by corollary, for real quadratic fields – is obtained at s=1. Finally, we discuss two applications related to Hilbert's 12th problem: numerical computation of Stark units in the rank 1 real quadratic case, and computation of fiducial vectors of Heisenberg SIC-POVMs.
*Advisors/Committee Members: Lagarias, Jeffrey C (committee member), Doering, Charles R (committee member), Koch, Sarah Colleen (committee member), Prasanna, Kartik (committee member), Snowden, Andrew (committee member), Zieve, Michael E (committee member).*

Subjects/Keywords: number theory; indefinite theta function; zeta function; real quadratic field; Kronecker limit formula; SIC-POVM; Mathematics; Science

…in the *real* *field* and
the elliptic modular function in the imaginary *quadratic* number *field*… …formula). Let K be a *real* *quadratic* *field*,
and let A be a narrow ray ideal class modulo f… …of the ring of integers of a *real* *quadratic* number *field* K with the property that,
if ε… …for rank 1 “Stark units” over a *real* *quadratic* base *field*. It deals with the
same cases as… …rational number or of the imaginary *quadratic* *field*,
any algebraic *field* whatever is laid down of…

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APA (6^{th} Edition):

Kopp, G. (2017). Indefinite Theta Functions and Zeta Functions. (Doctoral Dissertation). University of Michigan. Retrieved from http://hdl.handle.net/2027.42/140957

Chicago Manual of Style (16^{th} Edition):

Kopp, Gene. “Indefinite Theta Functions and Zeta Functions.” 2017. Doctoral Dissertation, University of Michigan. Accessed August 04, 2020. http://hdl.handle.net/2027.42/140957.

MLA Handbook (7^{th} Edition):

Kopp, Gene. “Indefinite Theta Functions and Zeta Functions.” 2017. Web. 04 Aug 2020.

Vancouver:

Kopp G. Indefinite Theta Functions and Zeta Functions. [Internet] [Doctoral dissertation]. University of Michigan; 2017. [cited 2020 Aug 04]. Available from: http://hdl.handle.net/2027.42/140957.

Council of Science Editors:

Kopp G. Indefinite Theta Functions and Zeta Functions. [Doctoral Dissertation]. University of Michigan; 2017. Available from: http://hdl.handle.net/2027.42/140957