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Baylor University

1.
Streit, Brian Quantz.
Conformal mapping methods for *spectral* *zeta* function calculations.

Degree: PhD, Baylor University. Dept. of Mathematics., 2015, Baylor University

URL: http://hdl.handle.net/2104/9573

We first show how to relate two spectral zeta functions corresponding to conformally equivalent two-dimensional smooth Riemannian manifolds. Next, the functional determinant of the Laplacian on an annulus is used to calculate the functional determinant of the Laplacian on a region bounded by two ellipses. We develop perturbation theory for Hermitian partial differential operators and show how this, combined with a conformal map from a disk to an elliptic region, can be used to derive a perturbative expansion for the spectral zeta function of the Laplacian on an elliptic region that is nearly circular. Finally, this perturbative expansion of the zeta function is used to approximate quantities of interest such as the functional determinant and heat kernel coefficients.
*Advisors/Committee Members: Kirsten, Klaus, 1962- (advisor).*

Subjects/Keywords: Conformal. Map. Spectral. Zeta. Functions.

Record Details Similar Records

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

APA (6^{th} Edition):

Streit, B. Q. (2015). Conformal mapping methods for spectral zeta function calculations. (Doctoral Dissertation). Baylor University. Retrieved from http://hdl.handle.net/2104/9573

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

Streit, Brian Quantz. “Conformal mapping methods for spectral zeta function calculations.” 2015. Doctoral Dissertation, Baylor University. Accessed January 24, 2021. http://hdl.handle.net/2104/9573.

MLA Handbook (7^{th} Edition):

Streit, Brian Quantz. “Conformal mapping methods for spectral zeta function calculations.” 2015. Web. 24 Jan 2021.

Vancouver:

Streit BQ. Conformal mapping methods for spectral zeta function calculations. [Internet] [Doctoral dissertation]. Baylor University; 2015. [cited 2021 Jan 24]. Available from: http://hdl.handle.net/2104/9573.

Council of Science Editors:

Streit BQ. Conformal mapping methods for spectral zeta function calculations. [Doctoral Dissertation]. Baylor University; 2015. Available from: http://hdl.handle.net/2104/9573

Brigham Young University

2.
Wu, Dongsheng.
Eigenvalues of Differential Operators and Nontrivial Zeros of L-* functions*.

Degree: PhD, 2020, Brigham Young University

URL: https://scholarsarchive.byu.edu/cgi/viewcontent.cgi?article=9729&context=etd

The Hilbert-P\'olya conjecture asserts that the non-trivial zeros of the Riemann zeta function ζ(s) correspond (in a certain canonical way) to the eigenvalues of some positive operator. R. Meyer constructed a differential operator D_{-} acting on a function space \H and showed that the eigenvalues of the adjoint of D_{-} are exactly the nontrivial zeros of ζ(s) with multiplicity correspondence.
We follow Meyer's construction with a slight modification. Specifically, we define two function spaces \H_\cap and \H_{-} on (0,∞) and characterize them via the Mellin transform. This allows us to show that Z\H_\cap\subseteq\H_{-} where Zf(x)=∑_{n=1}^∞ f(nx). Also, the differential operator D given by Df(x)=-xf'(x) induces an operator D_{-} on the quotient space \H=\H_{-}/Z\H_\cap. We show that the eigenvalues of D_{-} on \H are exactly the nontrivial zeros of ζ(s). Moreover, the geometric multiplicity of each eigenvalue is one and the algebraic multiplicity of each eigenvalue is its vanishing order as a nontrivial zero of ζ(s).
We generalize our construction on the Riemann zeta function to some L-functions, including the Dirichlet L-functions and L-functions associated with newforms in 𝓢_{k}(Γ_{0}(M)) with M ≥ 1 and k being a positive even integer. We give spectral interpretations for these L-functions in a similar fashion.

Subjects/Keywords: Hilbert P\'olya conjecture; Riemann zeta function; $L$-functions; spectral interpretation; Poisson summation formulass; Physical Sciences and Mathematics

Record Details Similar Records

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

APA (6^{th} Edition):

Wu, D. (2020). Eigenvalues of Differential Operators and Nontrivial Zeros of L-functions. (Doctoral Dissertation). Brigham Young University. Retrieved from https://scholarsarchive.byu.edu/cgi/viewcontent.cgi?article=9729&context=etd

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

Wu, Dongsheng. “Eigenvalues of Differential Operators and Nontrivial Zeros of L-functions.” 2020. Doctoral Dissertation, Brigham Young University. Accessed January 24, 2021. https://scholarsarchive.byu.edu/cgi/viewcontent.cgi?article=9729&context=etd.

MLA Handbook (7^{th} Edition):

Wu, Dongsheng. “Eigenvalues of Differential Operators and Nontrivial Zeros of L-functions.” 2020. Web. 24 Jan 2021.

Vancouver:

Wu D. Eigenvalues of Differential Operators and Nontrivial Zeros of L-functions. [Internet] [Doctoral dissertation]. Brigham Young University; 2020. [cited 2021 Jan 24]. Available from: https://scholarsarchive.byu.edu/cgi/viewcontent.cgi?article=9729&context=etd.

Council of Science Editors:

Wu D. Eigenvalues of Differential Operators and Nontrivial Zeros of L-functions. [Doctoral Dissertation]. Brigham Young University; 2020. Available from: https://scholarsarchive.byu.edu/cgi/viewcontent.cgi?article=9729&context=etd

3.
Lal, Nishu.
*Spectral**Zeta* *Functions* of Laplacians on Self-Similar Fractals.

Degree: Mathematics, 2012, University of California – Riverside

URL: http://www.escholarship.org/uc/item/888903d2

This thesis investigates the spectral zeta function of fractal differential operators such as the Laplacian on the unbounded (i.e., infinite) Sierpinski gasket and a self-similar Sturm – Liouville operator associated with a fractal self-similar measure on the half-line. In the latter case, C. Sabot discovered the relation between the spectrum of this operator and the iteration of a rational map of several complex variables, called the renormalization map. We obtain a factorization of the spectral zeta function of such an operator, expressed in terms of the Dirac delta hyperfunction, a geometric zeta function, and the zeta function associated with the dynamics of the corresponding renormalization map, viewed either as a polynomial function on the complex plane (in the first case) or (in the second case) as a polynomial on the complex projective plane. Our first main result extends to the case of the fractal Laplacian on the unbounded Sierpinski gasket a factorization formula obtained by M. Lapidus for the spectral zeta function of a fractal string and later extended by A. Teplyaev to the bounded (i.e., finite) Sierpinski gasket and some other decimable fractals. Furthermore, our second main result generalizes these factorization formulas to the renormalization maps of several complex variables associated with fractal Sturm – Liouville operators. Moreover, as a corollary, in the very special case when the underlying self-similar measure is Lebesgue measure on [0, 1], we obtain a representation of the Riemann zeta function in terms of the dynamics of a certain polynomial on the complex projective plane, thereby extending to several variables an analogous result by A. Teplyaev.

Subjects/Keywords: Mathematics; Analysis on fractals; decimation method; Dirac delta hyperfunction; fractal Sturm-Liouville operators; multivariable complex dynamics; spectral zeta functions

…ABSTRACT OF THE DISSERTATION
*Spectral* *Zeta* *Functions* of Laplacians on Self-Similar… …study of *spectral* *zeta* *functions* on fractals is inspired
by M. Lapidus [22, 27] in… …the *spectral* *zeta* function of fractal differential operators such
as the Laplacian on the… …the *spectral* *zeta* function of such an operator,
expressed in terms of the Dirac delta… …unbounded Sierpinski gasket a factorization
formula obtained by M. Lapidus for the *spectral* *zeta*…

Record Details Similar Records

❌

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

APA (6^{th} Edition):

Lal, N. (2012). Spectral Zeta Functions of Laplacians on Self-Similar Fractals. (Thesis). University of California – Riverside. Retrieved from http://www.escholarship.org/uc/item/888903d2

Note: this citation may be lacking information needed for this citation format:

Not specified: Masters Thesis or Doctoral Dissertation

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

Lal, Nishu. “Spectral Zeta Functions of Laplacians on Self-Similar Fractals.” 2012. Thesis, University of California – Riverside. Accessed January 24, 2021. http://www.escholarship.org/uc/item/888903d2.

Note: this citation may be lacking information needed for this citation format:

Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7^{th} Edition):

Lal, Nishu. “Spectral Zeta Functions of Laplacians on Self-Similar Fractals.” 2012. Web. 24 Jan 2021.

Vancouver:

Lal N. Spectral Zeta Functions of Laplacians on Self-Similar Fractals. [Internet] [Thesis]. University of California – Riverside; 2012. [cited 2021 Jan 24]. Available from: http://www.escholarship.org/uc/item/888903d2.

Note: this citation may be lacking information needed for this citation format:

Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

Lal N. Spectral Zeta Functions of Laplacians on Self-Similar Fractals. [Thesis]. University of California – Riverside; 2012. Available from: http://www.escholarship.org/uc/item/888903d2

Not specified: Masters Thesis or Doctoral Dissertation