On the mean square of quadratic Dirichlet L-functions at 1
by
Henri Virtanen
University of Turku, Finland

We will briefly outline the proof for the following mean square estimate for the quadratic Dirichlet L-function at s=1;

å c  L2(1, c) = AX + P(X)X1/2 + O(X1/2w(X)),

where the sum is taken over all real primitive non-principal characters with conductor at most X. The constant A and the function P(X), which is of the order logX, could be determine exactly. In the error term, the function w(X) is similar to that occuring in the error term of the prime number theorem, which tends to zero as X tends to infinity. This improves the earlier estimate with only one main term obtained by Jutila in 1973.

Date received: February 12, 2008