Teaching Documents by Kimberly Logan
Used Spring 2015 at Metro State for Math 110: Math for Liberal Arts
Syllabus for Math 120: Precalculus at Metro State.
Outline of guidelines for Math 110 project.
Papers by Kimberly Logan
We prove that all the zeros of a certain family of meromorphic functions are on the critical line... more We prove that all the zeros of a certain family of meromorphic functions are on the critical line Re(s) = 1/2, and are simple (except possibly when s = 1/2). We prove this by relating the zeros to the discrete spectrum of unbounded self-adjoint operators. For example, for h(s) a meromorphic function with no zeros in Re(s) > 1/2, real-valued on R, and h(1−s) h(s) |s| 1− in Re(s) > 1/2, the only zeros of h(s) ± h(1 − s) are on the critical line. More generally, we can allow certain patterns of finitely-many zeros and poles in the right half-plane. One instance of such a function h is h(s) = ξ k (2s), the completed zeta-function of a number field k, or, more generally, many self-dual automorphic L-functions. We use spectral theory suggested by results of Lax-Phillips and ColinDeVerdière. This simplifies ideas of W.
Physicists such as Green, Vanhove, et al show that differential equations involving automorphic f... more Physicists such as Green, Vanhove, et al show that differential equations involving automorphic forms govern the behavior of gravitons. One particular point of interest is solutions to (∆−λ)u = E α E β on an arithmetic quotient of the exceptional group E 8 . We establish that the existence of a solution to (∆ − λ)u = E α E β on the simpler space SL 2 (Z)\SL 2 (R) for certain values of α and β depends on nontrivial zeros of the Riemann zeta function ζ(s). Further, when such a solution exists, we use spectral theory to solve (∆ − λ)u = E α E β on SL 2 (Z)\SL 2 (R) and provide proof of the meromorphic continuation of the solution. The construction of such a solution uses Arthur truncation, the Maass-Selberg formula, and automorphic Sobolev spaces. 7
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Teaching Documents by Kimberly Logan
Papers by Kimberly Logan