A hybrid Euler-Hadamard product for the Riemann zeta function

Abstract

We use a smoothed version of the explicit formula to find an approximation to the Riemann zeta function as a product over its nontrivial zeros multiplied by a product over the primes. We model the first product by characteristic polynomials of random matrices. This provides a statistical model of the zeta function that involves the primes in a natural way. We then employ the model in a heuristic calculation of the moments of the modulus of the zeta function on the critical line. This calculation illuminates recent conjectures for these moments based on connections with random matrix theory.

Moments of moments of characteristic polynomials of random unitary matrices and lattice point counts

“…The h = 0 case is from the work of Keating and Snaith [45]. Moreover, for an explanation in the case h = 0 of why the moments factor asymptotically into an arithmetic term a(s) and a random matrix term F(s, 0), see [36].…”

Freezing transitions and extreme values: random matrix theory, and disordered landscapes

“…It has since been the subject of a number of studies (e.g. [9][10][11]23]), but we are still far from a complete understanding. For example, the role played by arithmetic is still being elucidated, although the hybrid model of [10] suggests that at leading order this contribution decouples from the random-matrix component.…”

“…The calculations outlined in the previous section are based on estimating the asymptotics of the moments of the characteristic polynomials, and using the high moments to determine the extreme values. The connections between these moments and those of the zeta function are now relatively well understood, at least conjecturally [7,9,10,23]. For example, to leading order as T → ∞ with N = log(T/2π), where a(λ) is defined by (2.26).…”

“…Over the past decade it has become a well-established paradigm that many statistical properties of the Riemann zeta function along the critical line can be understood by comparing them to analogous properties of the characteristic polynomials of random matrices [7][8][9][10][11]. Let U N be an N × N unitary matrix, chosen uniformly at random from the unitary group U(N) (i.e.…”

Untitled

“…To date, for suitably restricted test functions the statistics of zeros of many natural families of L-functions have been shown to agree with statistics of eigenvalues of matrices from the classical compact groups, including Dirichlet L-functions, elliptic curves, cuspidal newforms, Maass forms, number field L-functions, and symmetric powers of GL 2 automorphic representations Dueñez and Miller 2006;Fouvry and Iwaniec 2003;Gao 2005;Güloglu 2005;Hughes and Miller 2007;Hughes and Rudnick 2003;Iwaniec et al 2000;Katz and Sarnak 1999a;1999b;Miller 2004;Miller and Peckner 2012;Ricotta and Royer 2011;Royer 2001;Rubinstein 2001;Shin and Templier 2012;Yang 2009;Young 2006], to name a few, as well as nonsimple families formed by Rankin-Selberg convolution [Dueñez and Miller 2009]. In addition to predicting the main term (see for example [Conrey 2001;Katz and Sarnak 1999a;1999b;Keating and Snaith 2000a;2000b;), techniques from random matrix theory have led to models that capture the lower order terms in their full arithmetic glory for many families of L-functions (see for example the moment conjectures in [Conrey et al 2005a] or the hybrid model in [Gonek et al 2007]). Since the main terms agree with either unitary, symplectic or orthogonal symmetry, it is only in the lower order terms that we can break this universality and see the arithmetic of the family enter.…”