Sharp Upper Bounds for Fractional Moments of the Riemann Zeta Function

Abstract

We prove lower bounds for the discrete negative 2kth moment of the derivative of the Riemann zeta function for all fractional k 0. The bounds are in line with a conjecture of Gonek and Hejhal. Along the way, we prove a general formula for the discrete twisted second moment of the Riemann zeta function. This agrees with a conjecture of Conrey and Snaith.when k is fractional. These can give information about small values of the derivative ζ ′ (ρ), and in the special case k = 1/2 are related to partial sums of the Möbius function (e.g. see Theorem 14.27 of [24]). A natural assumption when considering these moments is that all the non-trivial zeros of the zeta function are simple. We therefore assume this throughout the paper unless otherwise mentioned.Gonek [9] and Hejhal [11] independently conjectured thatfor all k 0. However, the range of k in which this conjecture holds seems to be in doubt since Gonek (unpublished) has suggested that there exist infinitely many zeros ρ for which ζ ′ (ρ) −1 ≫ γ 1/3−ǫ , in which case the conjecture would fail for k > 3/2. Hughes, Keating and O'Connell [12] used random matrix theory to predict a precise constant in this conjecture. Interestingly, their formulas on the random matrix theory side undergo a phase change at the point k = 3/2 which gives alternative evidence to suggest that the conjecture fails for k > 3/2. Aside from these conjectures, little is known about J −k (T ). Assuming the Riemann hypothesis (RH) Gonek [9] showed thatand that by applying Hölder's inequalityfor all k > 0. Milinovich and Ng [16] later refined Gonek's bound by showing thatThe value of the constant here is half the conjectured value [9,12]. In the special case k = 1/2, Heath-Brown [24, pg. 386], shows J −1/2 (T ) ≫ T via the connection with n x µ(n) on assuming RH. Our aim in this paper is to improve these lower bounds.

Moments of the Hurwitz zeta function on the critical line

“…The upper bound was known unconditionally for k = 1/n, n ∈ N due to Heath-Brown [HB81], and for k = 1 + 1/n, n ∈ N due to Bettin, Chandee and Radziwi l l [BCR17]. Recently, Heap, Radziwi l l and Soundararajan [HRS19] subsumed both of these results by proving the upper bound unconditionally for 0 ≤ k ≤ 2.…”

Bounds for moments of cubic and quartic Dirichlet $L$-functions

“…In [22], M. Radziwi l l and K. Soundararajan developed an upper bounds principle to study moments of families of L-functions unconditionally and applied their principal for the family of quadratic twists of elliptic L-functions. The principal was carried out by W. Heap, M. Radziwi l l and K. Soundararajan in [13] to establish sharp upper bounds for M k (T ) for 0 ≤ k ≤ 2 unconditionally. A dual principle was developed by W. Heap and K. Soundararajan in [14] to establish sharp lower bounds for M k (T ) for all real k ≥ 0 unconditionally.…”

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

“…Under RH, M. B. Milinovich [22] established essentially upper bounds of the correct order of magnitude for I k,l (T ) for positive integers k, l. His result was further improved to yield optimal upper bounds by A. Ivić [19] for I k,2 (T ) for positive integers k. The methods employed in [22] and [19] allow one to deduce upper bounds for I k,l (T ) from the corresponding ones for M k (T ). As sharp upper bounds for M k (T ) are known for all k ≥ 0 under RH from the work of K. Soundararajan [35] and for all 0 ≤ k ≤ 2 unconditionally from the work of W. Heap, M. Radziwi l l and K. Soundararajan [15], we may apply the methods in [22] and [19] to derive that unconditionally for 1/2 ≤ k ≤ 2 and under RH for k ≥ 2, we have for all integers l ≥ 1,…”