Some remarks on the mean value of the Riemann zetafunction and other Dirichlet series. III

Fractional moments of Dirichlet L-functions

“…with δ > 0 to be specified later, see (6) and (7). We emphasize that, for the rest of this paper, all constants implied by the Vinogradov symbol will be uniform in σ for the ranges specified.…”

“…with positive non-integral values of k, have been investigated by a number of authors, including Ramachandra [5], [6], Conrey and Ghosh [1] and HeathBrown [3]. In particular the above papers by Ramachandra show, under the Riemann Hypothesis, that…”

Lower bounds of discrete moments of the derivatives of the Riemann zeta-function on the critical line

“…the conditional one by Ramachandra [16], we can deduce that holds for any rational k ≥ 0, resp. under the assumption of the Riemann hypothesis for any real k ≥ 0.…”

“…Furthermore, we must note that in the case l = 0, the bounds of Corollary 1.1 were proved by Heath-Brown [7] for any positive rational k and under the assumption of the Riemann hypothesis by Ramachandra [16] for any positive real k.…”

Lower bounds for moments of automorphic $L$-functions over short intervals

“…For the Riemann zeta-function ζ(s), a conjecture states that the 2k-th moment should be asymptotic to C k T (log T ) . Actually, this latter result was proved by Ramachandra [11] for all positive integers k, by Heath-Brown [4] for all positive rational numbers k, and under the Riemann Hypothesis by Ramachandra [10] for all positive real numbers k. In [1] the authors propose conjectures for the full asymptotics of the moments of general L-functions. In particular, the paper provides conjectures for the moments of the Riemann zeta-function, the family of primitive Dirichlet L-functions, quadratic twists of L-functions, and automorphic L-functions attached to cusp forms.…”