On Some Weighted Average Values of L-functions

Let q ≥ 2 and N ≥ 1 be integers. W. Zhang recently proved that for any fixed ε > 0 and q ≤ N ≤ q1/2−ε , ∑ χ 6=χ0 ∣∣∣∣ N ∑ n=1 χ(n) ∣∣∣∣2|L(1, χ)|2 = (1+ o(1))αq q N , where the sum is taken over all nonprincipal characters χ modulo q, L(1, χ) denotes the L-functions corresponding to χ , and αq = qo(1) is some explicit function of q. Here we improve this result and show that the same asymptotic formula holds in the essentially full range q ≤ N ≤ q1−ε . 2000 Mathematics subject classification: primary 11M06.