We investigate the zeros of Epstein zeta functions associated with a positive definite quadratic form with rational coefficients. Davenport and Heilbronn, and also Voronin, proved the existence of zeros of Epstein zeta functions off the critical line when the class number of the quadratic form is bigger than 1. These authors give lower bounds for the number of zeros in strips that are of the same order as the more easily proved upper bounds. In this paper, we improve their results by providing asymptotic formulas for the number of zeros.
This paper is part of my Ph.D. thesis at Yonsei University. I would like to express my gratitude to my advisor Professor Haseo Ki for his unfailing support. I really appreciate Professor Steve Gonek for helping me to revise this paper. I thank Professor Enrico Bombieri for his kindness and many invaluable suggestions and corrections to this paper, Professor Roger Heath-Brown for showing me his interest and suggestions for this paper, and the reviewer for his or her careful reading and many comments.
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