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University of Cambridge

1. Chen, Cangxiong. ON ASAI’S FUNCTION ANALOGOUS TO log |η(z)|.

Degree: PhD, 2015, University of Cambridge

Kronecker’s first limit formula describes the constant term in the Laurent expansion of a non-holomorphic Eisenstein series at one of its poles. Asai generalised the limit formula to Eisenstein series of level one defined for a number field with class number one and obtained a function analogous to the logarithm of the absolute value of the eta function. In this thesis we reformulate Asai’s function adelically using the theory of admissible representations for GL2 and simultaneously remove the restriction on class number and level. As an application of the method, we give explicit computations of the Rankin-Selberg integral with two Eisenstein series and a cusp form.

Subjects/Keywords: Algebraic number theory; Automorphic forms; Asai's function; Eisenstein series; Kronecker Limit Formula; L-functions; Rankin-Selberg integral

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APA (6^{th} Edition):

Chen, C. (2015). ON ASAI’S FUNCTION ANALOGOUS TO log |η(z)|. (Doctoral Dissertation). University of Cambridge. Retrieved from https://www.repository.cam.ac.uk/handle/1810/306109https://www.repository.cam.ac.uk/bitstream/1810/306109/2/license.txt

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

Chen, Cangxiong. “ON ASAI’S FUNCTION ANALOGOUS TO log |η(z)|.” 2015. Doctoral Dissertation, University of Cambridge. Accessed August 04, 2020. https://www.repository.cam.ac.uk/handle/1810/306109https://www.repository.cam.ac.uk/bitstream/1810/306109/2/license.txt.

MLA Handbook (7^{th} Edition):

Chen, Cangxiong. “ON ASAI’S FUNCTION ANALOGOUS TO log |η(z)|.” 2015. Web. 04 Aug 2020.

Vancouver:

Chen C. ON ASAI’S FUNCTION ANALOGOUS TO log |η(z)|. [Internet] [Doctoral dissertation]. University of Cambridge; 2015. [cited 2020 Aug 04]. Available from: https://www.repository.cam.ac.uk/handle/1810/306109https://www.repository.cam.ac.uk/bitstream/1810/306109/2/license.txt.

Council of Science Editors:

Chen C. ON ASAI’S FUNCTION ANALOGOUS TO log |η(z)|. [Doctoral Dissertation]. University of Cambridge; 2015. Available from: https://www.repository.cam.ac.uk/handle/1810/306109https://www.repository.cam.ac.uk/bitstream/1810/306109/2/license.txt

2. 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

…sending s → 1 in this Fourier expansion, we obtain the first *Kronecker* *limit*
*formula*.
(1.10… …*Kronecker* *limit* *formula* may be found
in [29], chapter 20, pages 273–275.
From this… …mathematicians have found analogues of the *Kronecker* *limit* *formula* in
other settings. With an eye… …fields beyond the imaginary quadratic case.
Hecke found a *Kronecker* *limit* *formula* for real… …a paper of Zagier [51]. A *Kronecker*
*limit* *formula* for narrow (modulus 1…

Record Details Similar Records

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

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