Analytic Number Theory, Complex Variable and Supercomputers1
نویسندگان
چکیده
This paper is the nal report of an undergraduate honors thesis project advised by Prof. Dennis Hejhal of the School of Mathematics, University of Minnesota. The main purpose of this project is to examine the analytic properties of certain \quantum-mechanical particles" in Lobachevsky space. The results were obtained on vectorized CRAY serial supercomputers, a CRAY 64-CPU T3D massively parallel system, and a 1K-CPU massively parallel system CM-5 located in University of Minnesota. Using complex arithmetic, we have successfully determined numerous Fourier coe cients for certain types of holomorphic modular forms, including the Ramanujan -function. Our experiments involve both arithmetic and non-arithmetic groups. The treatment of the latter is new. Analyzing the output data enables us to experimentally justify a number of properties. Finally, a veri cation of a Central Limit Theorem for automorphic functions on Hecke Groups was attempted, and very promising results have been obtained. Analytic Number Theory, Complex Variable and Supercomputers
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