Real zeros of Dedekind zeta functions of real quadratic fields

Let χ be a primitive, real and even Dirichlet character with conductor q, and let s be a positive real number. An old result of H. Davenport is that the cycle sums Sν(s, χ) = ∑(ν+1)q−1 n=νq+1 χ(n) ns , ν = 0, 1, 2, . . . , are all positive at s = 1, and this has the immediate important consequence of the positivity of L(1, χ). We extend Davenport’s idea to show that in fact for ν ≥ 1, Sν(s, χ) > 0 for all s with 1/2 ≤ s ≤ 1 so that one can deduce the positivity of L(s, χ) by the nonnegativity of a finite sum ∑t ν=0 Sν(s, χ) for any t ≥ 0. A simple algorithm then allows us to prove numerically that L(s, χ) has no positive real zero for a conductor q up to 200,000, extending the previous record of 986 due to Rosser more than 50 years ago. We also derive various estimates explicit in q of the Sν(s, χ) as well as the shifted cycle sums Tν(s, χ) := ∑(ν+1)q+ q/2 n=νq+ q/2 +1 χ(n) ns considered previously by Leu and Li for s = 1. These explicit estimates are all rather tight and may have independent interests.