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On the zeros of Epstein zeta functions
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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.
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The Epstein zeta-function contains a positive proportion of non-trivial zeros on the critical line
For any real linear combination of distinct Hecke L-functions over an imaginary quadratic field, including Epstein zeta-functions of binary quadratic forms, a positive proportion of non-trivial zeros lies on Re s = 1/2.
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