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Universiteit Utrecht

1.
Zwegers, S.P.
Mock *Theta* Functions.

Degree: 2002, Universiteit Utrecht

URL: http://dspace.library.uu.nl:8080/handle/1874/878

The mock theta functions were invented by the Indian mathematician Srinivasa Ramanujan, who lived from 1887 until 1920. He discovered them shortly before his death. In this dissertation, I consider several of the examples that Ramanujan gave of mock theta functions, and relate them to real-analytic modular forms of weight 1/2.
In Chapter 1, I consider a sum, which was also studied by Lerch. This Lerch sum transforms almost as a Jacobi form under substitutions in (upsilon, nu, tau ). I show that the transformation behaviour becomes that of a Jacobi form if we add a (relatively simple) correction term. This correction term is real-analytic in (upsilon, nu, tau) but not holomorphic. For special values of (upsilon, nu), we could call the Lerch sum (considered as a function of tau ) a mock theta function, although these examples were not considered by Ramanujan.
In Chapter 2, I consider theta functions for indefinite quadratic forms. These indefinite theta functions are modified versions of theta functions introduced by Göttsche and Zagier. I find elliptic and modular transformation properties of these functions, similar to the properties of theta functions associated to positive definite quadratic forms. In the case of positive definite quadratic forms, the theta functions are holomorphic. The theta functions in the chapter are not holomorphic. By taking special values of certain parameters, we get most of the examples of mock theta functions given by Ramanujan.
Andrews gave most of the fifth order mock theta functions as Fourier coefficients of meromorphic Jacobi forms, namely certain quotients of ordinary Jacobi theta-series. This is the motivation for the study of the modularity of Fourier coefficients of meromorphic Jacobi forms, in Chapter 3. We find that modularity follows on adding a real-analytic correction term to the Fourier coefficients.
In Chapter 4, I use the results from Chapter 2 to get the modular transformation properties of the seventh-order mock v-functions and most of the fifth-order functions. The final result is that we can write each of these mock theta-functions as the sum of two functions ? and G, where:
- ? is a real-analytic modular form of weight 1/2 and is an eigenfunction of the appropriate Casimir operator with eigenvalue 3/16 (this is also the eigenvalue of holomorphic modular forms of this weight); and
- G is a theta series associated to a negative definite unary quadratic form. Moreover G is bounded as ? tends vertically to any rational limit.
Many of the results of Chapter 4 could be deduced using the methods from Chapter 1 or Chapter 3 instead of Chapter 2. This means that I have actually given three approaches to proving modularity properties of the mock theta -functions.

Subjects/Keywords: Wiskunde en Informatica; mock theta function; indefinite theta function; indefinite quadratic form; theta series; Jacobi form; real-analytic modular form

Record Details Similar Records

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

APA (6^{th} Edition):

Zwegers, S. P. (2002). Mock Theta Functions. (Doctoral Dissertation). Universiteit Utrecht. Retrieved from http://dspace.library.uu.nl:8080/handle/1874/878

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

Zwegers, S P. “Mock Theta Functions.” 2002. Doctoral Dissertation, Universiteit Utrecht. Accessed August 12, 2020. http://dspace.library.uu.nl:8080/handle/1874/878.

MLA Handbook (7^{th} Edition):

Zwegers, S P. “Mock Theta Functions.” 2002. Web. 12 Aug 2020.

Vancouver:

Zwegers SP. Mock Theta Functions. [Internet] [Doctoral dissertation]. Universiteit Utrecht; 2002. [cited 2020 Aug 12]. Available from: http://dspace.library.uu.nl:8080/handle/1874/878.

Council of Science Editors:

Zwegers SP. Mock Theta Functions. [Doctoral Dissertation]. Universiteit Utrecht; 2002. Available from: http://dspace.library.uu.nl:8080/handle/1874/878

2.
Zwegers, S.P.
Mock *Theta* Functions.

Degree: 2002, University Utrecht

URL: https://dspace.library.uu.nl/handle/1874/878 ; URN:NBN:NL:UI:10-1874-878 ; URN:NBN:NL:UI:10-1874-878 ; https://dspace.library.uu.nl/handle/1874/878

The mock theta functions were invented by the Indian mathematician Srinivasa Ramanujan, who lived from 1887 until 1920. He discovered them shortly before his death. In this dissertation, I consider several of the examples that Ramanujan gave of mock theta functions, and relate them to real-analytic modular forms of weight 1/2.
In Chapter 1, I consider a sum, which was also studied by Lerch. This Lerch sum transforms almost as a Jacobi form under substitutions in (upsilon, nu, tau ). I show that the transformation behaviour becomes that of a Jacobi form if we add a (relatively simple) correction term. This correction term is real-analytic in (upsilon, nu, tau) but not holomorphic. For special values of (upsilon, nu), we could call the Lerch sum (considered as a function of tau ) a mock theta function, although these examples were not considered by Ramanujan.
In Chapter 2, I consider theta functions for indefinite quadratic forms. These indefinite theta functions are modified versions of theta functions introduced by Göttsche and Zagier. I find elliptic and modular transformation properties of these functions, similar to the properties of theta functions associated to positive definite quadratic forms. In the case of positive definite quadratic forms, the theta functions are holomorphic. The theta functions in the chapter are not holomorphic. By taking special values of certain parameters, we get most of the examples of mock theta functions given by Ramanujan.
Andrews gave most of the fifth order mock theta functions as Fourier coefficients of meromorphic Jacobi forms, namely certain quotients of ordinary Jacobi theta-series. This is the motivation for the study of the modularity of Fourier coefficients of meromorphic Jacobi forms, in Chapter 3. We find that modularity follows on adding a real-analytic correction term to the Fourier coefficients.
In Chapter 4, I use the results from Chapter 2 to get the modular transformation properties of the seventh-order mock v-functions and most of the fifth-order functions. The final result is that we can write each of these mock theta-functions as the sum of two functions ? and G, where:
- ? is a real-analytic modular form of weight 1/2 and is an eigenfunction of the appropriate Casimir operator with eigenvalue 3/16 (this is also the eigenvalue of holomorphic modular forms of this weight); and
- G is a theta series associated to a negative definite unary quadratic form. Moreover G is bounded as ? tends vertically to any rational limit.
Many of the results of Chapter 4 could be deduced using the methods from Chapter 1 or Chapter 3 instead of Chapter 2. This means that I have actually given three approaches to proving modularity properties of the mock theta -functions.

Subjects/Keywords: mock theta function; indefinite theta function; indefinite quadratic form; theta series; Jacobi form; real-analytic modular form

Record Details Similar Records

❌

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

APA (6^{th} Edition):

Zwegers, S. P. (2002). Mock Theta Functions. (Doctoral Dissertation). University Utrecht. Retrieved from https://dspace.library.uu.nl/handle/1874/878 ; URN:NBN:NL:UI:10-1874-878 ; URN:NBN:NL:UI:10-1874-878 ; https://dspace.library.uu.nl/handle/1874/878

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

Zwegers, S P. “Mock Theta Functions.” 2002. Doctoral Dissertation, University Utrecht. Accessed August 12, 2020. https://dspace.library.uu.nl/handle/1874/878 ; URN:NBN:NL:UI:10-1874-878 ; URN:NBN:NL:UI:10-1874-878 ; https://dspace.library.uu.nl/handle/1874/878.

MLA Handbook (7^{th} Edition):

Zwegers, S P. “Mock Theta Functions.” 2002. Web. 12 Aug 2020.

Vancouver:

Zwegers SP. Mock Theta Functions. [Internet] [Doctoral dissertation]. University Utrecht; 2002. [cited 2020 Aug 12]. Available from: https://dspace.library.uu.nl/handle/1874/878 ; URN:NBN:NL:UI:10-1874-878 ; URN:NBN:NL:UI:10-1874-878 ; https://dspace.library.uu.nl/handle/1874/878.

Council of Science Editors:

Zwegers SP. Mock Theta Functions. [Doctoral Dissertation]. University Utrecht; 2002. Available from: https://dspace.library.uu.nl/handle/1874/878 ; URN:NBN:NL:UI:10-1874-878 ; URN:NBN:NL:UI:10-1874-878 ; https://dspace.library.uu.nl/handle/1874/878

3.
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

…variable coefficients used to define
an *indefinite* *theta* *function*.
Definition II.22. Let Ω = N… …1, 1); that is, Ω ∈ Hg . Define the *indefinite* *theta* *function*
c2
>
X
c
Im(… …*indefinite* zeta *function* using a Mellin transform of the *indefinite*
*theta* *function* with… …*function*.
13
Proposition II.24 (Elliptic transformation laws). The *indefinite* *theta*… …*indefinite* *theta* *function* satisfies the following transformation laws with respect to the Ω…

Record Details Similar Records

❌

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 12, 2020. http://hdl.handle.net/2027.42/140957.

MLA Handbook (7^{th} Edition):

Kopp, Gene. “Indefinite Theta Functions and Zeta Functions.” 2017. Web. 12 Aug 2020.

Vancouver:

Kopp G. Indefinite Theta Functions and Zeta Functions. [Internet] [Doctoral dissertation]. University of Michigan; 2017. [cited 2020 Aug 12]. 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