Asymptotics for Sums of Central Values of Canonical Hecke LSeries
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Let K = ℚ(√D) be an imaginary quadratic field of discriminant D with D > 3 and D = 3 mod 4. Let OK be the ring of integers of K, let ε be the quadratic character of K of conductor √DO K, and let ψk be a Hecke character of K of conductor √DOK satisfying ψk(αOK) = ε(α)α2k1, for (αOK√DO K) = 1, k ε ℤ≥1. Let h(D) be the class number of K and let εk be the set of h(D) Hecke characters of the form εk. If L(εk, s) denotes the Lseries of εk then its central value is L(εk, k). In our main theorem we establish for each even integer k ≥ 2 an asymptotic formula for the average 1/h(D) σεkεεk L(εk,K)/L((D),1) as D→∞. We then use this formula to prove that there exists an absolute constant λ ≥ 0 such that the number of nonvanishing central values in the family {L(εk, k) : εk ε εk} is ≫ Dλ as D→∞ © The Author 2007.
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