An Upper Bound for Discrete Moments of the Derivative of the Riemann Zeta‐function

Abstract

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

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

“…We shall instead apply the lower bounds principal of W. Heap and K. Soundararajan [9] in the proof of Theorem 1.1. The proof also uses the arguments by A. J. Harper in [8] and by S. Kirila in [12]. Combining Theorem 1.1 and the above mentioned result of S. Kirila in [12], we immediately obtain the following result concerning the order of magnitude of J k (T ).…”

“…One then expects to apply these approaches to obtain sharp bounds concerning J k (T ). In fact, it is pointed out in [12] that one should be able to establish sharp lower bounds for all real k > 0 using the approaches in [20,21]. The aim of this paper is to achieve this and our main result is as follows.…”

“…The cases when a = b of Lemma 2.1 are given in [12,Lemma 5.1] while the case a = b of Lemma 2.1 is trivial. Recall that the Riemann-von Mangoldt formula asserts (see [2,Chapter 15]) that…”

“…We follow the ideas of A. J. Harper in [8] and the notations of S. Kirila in [12] to define for a large number M depending on k only,…”

“…In [12], S. Kirila obtained sharp upper bounds for J k (T ) for all real k > 0 under RH. In particular, this implies the validity of (1.3) for natural numbers k without the extra ε power.…”