Moments of the Dedekind zeta function and other non-primitiveL-functions

Abstract

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.

Hybrid Euler-Hadamard product for quadratic Dirichlet L-functions in function fields

“…As there is only one pole of the integrand at u = 1 in the annulus between |u| = q −1+ε and |u| = q 1−ε , in view of (15) and (19) we conclude that…”

“…While in the original Keating and Snaith approach, the arithmetic factor was not predicted by the random matrix theory computation and had to be inserted in the formula in a somewhat ad‐hoc manner, the hybrid approach has the advantage of predicting the asymptotic formula in a more natural way. The hybrid Euler‐Hadamard product model has been extended to various cases .…”

Moments of the Hurwitz zeta function on the critical line

“…For the mean square of Z ℓ X (s), we use random matrix theory to model each L-function appearing in the product by random unitary matrices. One expects that the matrices representing distinct L-functions behave independently as in [Hea13,Conjecture 2]. This leads to Conjecture 1.6.…”

“…Such a hybrid Euler-Hadamard product was proved by Bui and Keating [BK07] in their study of moments in the q-aspect of Dirichlet Lfunctions at the central point s = 1/2 (see [BK07, Remark 1]). Similar hybrid Euler-Hadamard products have been used in the literature for studying moments in many other contexts such as for for orthogonal and symplectic families of L-functions [BK08]; for ζ ′ (s) [BGM15]; for the Dedekind zeta function ζ K (s) of a Galois extension K of Q [Hea13]; for quadratic Dirichlet L-functions over function fields [BF18], [AGK18]; for normalized symmetric square L-functions associated with SL 2 (Z) eigenforms [Dja13]; and for quadratic Dirichlet L-functions over function fields associated to irreducible polynomials [AS19]. With P (s, χ) and Z(s, χ) as in Theorem 1.3, we define P ℓ X (s) and Z ℓ X (s) by…”

On the Mertens Conjecture over Number Fields

“…Expanding upon this work, Hughes-Keating-O'Connell [HKO00] conjectured asymptotic formulae for all J k (T ), k > − 3 2 , giving explicit numerical constants depending on k. The random matrix approach has proven to produce reliable conjectures; for instance, the conjecture of Hughes-Keating-O'Connell in the case k = −1 agrees with that of Gonek. It is expected that similar heuristics would yield accurate conjectures for positive moments of Dedekind zeta functions as well (see [GHK07,BGM15,Hea21]). They would also be useful in support of (1.3).…”