Fractional moments of Dirichlet L-functions

Fractional moments of Dirichlet L-functions by D. R. Heath-Brown (Oxford) 1. Introduction.Mean-values of the type I k (T ) := T 0 |ζ(1/2 + it)| 2k dt, with positive non-integral values of k, have been investigated by a number of authors, including Ramachandra [5], [6], Conrey and Ghosh [1] and Heath-Brown [3].In particular the above papers by Ramachandra show, under the Riemann Hypothesis, thatfor all real k ≥ 0, and thatIt is natural to ask about the corresponding problem for Dirichlet Lfunctions in q-aspect, that is, to investigatefor positive real k.However rather little is known about this in general.The method of Rudnick and Soundararajan [7] enables one to show unconditionally that M k (q) k φ(q)(log q) k 2 for rational k ≥ 1, at least when q is prime.The method does not immediately apply to the case 0 < k < 1 and it would be interesting to establish lower bounds in this range.