The L q norm of a Dirichlet polynomial F (s) = N n=1 a n n −s is defined asthe Möbius function and d the divisor function. This result is used to prove that the L q norm of D N (s) := N n=1 n −1/2−s satisfies D N q ≫ (log N ) q/4 for 0 < q < ∞. By Helson's generalization of the M. Riesz theorem on the conjugation operator, the reverse inequality D N q ≪ (log N ) q/4 is shown to be valid in the range 1 < q < ∞. Similar bounds are found for a fairly large class of Dirichlet series including, on one of Selberg's conjectures, the Selberg class of L-functions.

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 However, the range of k in which this conjecture holds seems to be in doubt since Gonek (unpublished) has suggested that there exist infinitely many zeros ρ for which ζ ′ (ρ) −1 ≫ γ 1/3−ǫ , in which case the conjecture would fail for k 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.

Abstract. We give an asymptotic formula for the 2kth moment of a sum of multiplicative Steinhaus variables. This was recently computed independently by Harper, Nikeghbali and Radziwi l l. We also compute the 2kth moment of a truncated characteristic polynomial of a unitary matrix. This provides an asymptotic equivalence with the moments of Steinhaus variables. Similar results for multiplicative Rademacher variables are given.

This paper describes a method to compute lower bounds for moments of ζ$\zeta$ and L$L$‐functions. The method is illustrated in the case of moments of false|ζfalse(0false12+itfalse)false|$|\zeta (\tfrac{1}{2}+it)|$, where the results are new for small moments 0

We give a conjecture for the moments of the Dedekind zeta function of a Galois extension. This is achieved through the hybrid product method of Gonek, Hughes and Keating. The moments of the product over primes are evaluated using a theorem of Montgomery and Vaughan, whilst the moments of the product over zeros are conjectured using a heuristic method involving random matrix theory. The asymptotic formula of the latter is then proved for quadratic extensions in the lowest order case. We are also able to reproduce our moments conjecture in the case of quadratic extensions by using a modified version of the moments recipe of Conrey et al. Generalising our methods, we then provide a conjecture for moments of non-primitive L-functions, which is supported by some calculations based on Selberg’s conjectures.