Assuming the Riemann hypothesis, we establish an upper bound for the 2kth discrete moment of the derivative of the Riemann zeta-function at nontrivial zeros, where k is a positive real number. Our upper bound agrees with conjectures of Gonek and Hejhal and of Hughes, Keating and O'Connell. This sharpens a result of Milinovich. Our proof builds upon a method of Adam Harper concerning continuous moments of the zeta-function on the critical line. London under an exclusive licence. 2 . Hughes et al. inserted the product over primes here was inserted in an ad hoc manner, namely from a heuristic estimate for the case k = −1/2. Bui et al. [1] used a hybrid Euler-Hadamard product model for ζ (ρ) to suggest precisely where this product over primes comes from, essentially merging ideas from number theory and random matrix theory in the same way as Gonek et al. [11] did for moments of ζ ( 1 2 + it ). These conjectures remain open, but work has been done toward the implied upper and lower bounds conditionally on RH. Milinovich and Ng [17] obtained the expected lower boundfor any natural number k. In the other direction, Milinovich [16] showed thatfor any ε > 0. The purpose of this paper is to remove the ε in the exponent here and prove the following. THEOREM 1.1. Assume RH. Let k > 0. ThenTogether with (1.2), this shows thatCompleting the square then leads us to conclude that
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