Mean-value estimates for the derivative of the Riemann zeta-function.

摘要

Let ζ(s) denote the Riemann zeta-function. This thesis is concerned with estimating discrete moments of the form JkT= 1NT 0 0 arbitrary, there exist positive constants C1 = C1(k) and C2 = C2( k, ϵ) such that the inequalities C1logT kk+2≤Jk T≤C2 logTk k+2+3 hold when T is sufficiently large. The lower bound for Jk(T) was proved jointly with Nathan Ng.;Two related problems are also considered. Assuming the Riemann Hypothesis S. M. Gonek has shown that J1(T) ∼ 112logT 3 as T → ∞. As an application of the L-functions Ratios Conjectures, J.B. Conrey and N. Snaith made a precise conjecture for the lower-order terms in the asymptotic expression for J 1(T). By carefully following Gonek's original proof, we establish their conjecture.;The other problem is related to the average of the mean square of the reciprocal of ζ′(ρ). It is believed that the zeros of ζ(s) are all simple. If this is the case, then the sum Jk(T) is defined when k < 0 and, for certain small values of k, conjectures exist about its behavior. Assuming the Riemann Hypothesis and that the zeros of ζ(s) are simple, we establish a lower bound for J-1(T) that differs from the conjectured value by a factor of 2.