Asymptotics for Sums of Central Values of Canonical Hecke L-Series

Let K = (-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 be the quadratic character of K of conductor -DO K, and let k be a Hecke character of K of conductor -DOK satisfying k(OK) = ()2k-1, for (OK|-DO K) = 1, k 1. Let h(-D) be the class number of K and let k be the set of h(-D) Hecke characters of the form k. If L(k, s) denotes the L-series of k then its central value is L(k, k). In our main theorem we establish for each even integer k 2 an asymptotic formula for the average 1/h(-D) kk L(k,K)/L((-D),1) as D. We then use this formula to prove that there exists an absolute constant 0 such that the number of nonvanishing central values in the family {L(k, k) : k k} is D as D The Author 2007.

Asymptotics for Sums of Central Values of Canonical Hecke L-Series Academic Article