CONVOLUTIONS WITH PROBABILITY DISTRIBUTIONS, ZEROS OF L-FUNCTIONS, AND THE LEAST QUADRATIC NONRESIDUE

Let d be the density of a probability distribution that is compactly supported in the positive semi-axis. Under certain mild conditions we show that lim(x ->infinity) x Sigma(infinity)(n=1) d*(n)(x)/n = 1, where d*(n) := d * d * ... * d (sic) n times. We also show that if c > 0 is a given constant for which the function f (k) : = (d) over cap (k) 1 does not vanish on the line {k is an element of C : f k = -c}, where (d) over cap is the Fourier transform of d, then one has the asymptotic expansion Sigma(infinity)(n=1) d*(n)(x)/n = 1/x )1 + Sigma(k) m(k)e(-ikx) + O(e(-cx))) (x -> +infinity), where the sum is taken over those zeros k of f that lie in the strip {k is an element of C : -c < f k < 0}, m (k) is the multiplicity of any such zero, and the implied constant depends only on c. For a given distribution of this type, we briefly describe the location of the zeros k of f in the lower half-plane {k is an element of C : f k < 0}. For an odd prime p, let n(0)(p) be the least natural number such that (n vertical bar p) = 1, where (.vertical bar p) is the Legendre symbol. As an application of our work on probability distributions, we generalize a well known result of Heath-Brown concerning the exhibited behavior of the Dirichlet L-function L (s, (.vertical bar p)) under the assumption that the Burgess bound n(0)(p) << p(1/(4 root e)+epsilon) cannot be improved.