Sharp lower bounds for moments of $ζ'(ρ)$

Abstract

We study the 2k-th discrete moment of the derivative of the Riemann zeta-function at nontrivial zeros to establish sharp lower bounds for all real k ≥ 0 under the Riemann hypothesis (RH).

Lower bounds for moments of the derivative of the Riemann zeta function

“…We skip the proof of the above lemma as it can be established similar to those of [11,]. We deduce from the above lemma that in order to establish Theorem 1.1, it suffices to prove the following three propositions.…”

“…We note that a similar approach to the one used in [3, Section 5] has already been employed by M. B. Milinovich and N. Ng [23] in their study on lower bounds for the discrete moments of the derivative of ζ(s) at nontrivial zeros. Since these discrete moments can be regarded as analogues to I k,1 (T ) and are studied by the author in [11], the proof of Theorem 1.1 also makes use of some approaches there as well.…”

Lower bounds for negative moments of ζ′(ρ)$\zeta ^{\prime }(\rho )$

“…Much progress has since been made toward establishing (1.1) for the case 𝑘 ⩾ 0. In fact, based on the work in [4,14,16,18], we know that (1.1) is valid for all real 𝑘 ⩾ 0 on RH.…”