Integral Moments of L‐Functions

By J. Brian Conrey, David W. Farmer, Jon P Keating, Michael O. Rubinstein, N. Snaith. 2002

464 citations recorded by the source corpus.

We give a new heuristic for all of the main terms in the integral moments of various families of primitive L‐functions. The results agree with previous conjectures for the leading order terms. Our conjectures also have an almost identical form to exact expressions for the corresponding moments of the characteristic polynomials of either unitary, orthogonal, or symplectic matrices, where the moments are defined by the appropriate group averages. This lends support to the idea that arithmetical L‐functions have a spectral interpretation, and that their value distributions can be modelled using Random Matrix Theory. Numerical examples show good agreement with our conjectures. 2000 Mathematics Subject Classification 11M26, 15A52.

Subfields: Algebra and Number Theory, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/ff62981c70d0e91f8dc220b97a2bc2486b904374

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

In Mathematical Physics

Citations in the corpus

Autocorrelation of ratios of L-functions (2007, methodology): “We will use a modification of the recipe in [ 5 ]”
Moments of the Riemann zeta function (2006, methodology): “Analogous conjectures exist (see [2], [4], [11]) for moments of central values of L-functions in families, and in many cases lower bounds of the conjectured order are known (see [16] and [17]).”
The mean-square of Dirichlet L-functions (2007, background): “no terms between those of size φ∗(q) and the error-term of size q1/2+ǫ in contrast to Heath-Brown’s result. The point is that it is the primitive characters which form a family in the sense deﬁned in [CFKRS], and the set of all characters does not. In the case that qis prime, the set of all characte”
Families of L -Functions and Their Symmetry (2014, background): “In fact our notion of families provides a natural setting for the axiomatic recipies in [24], specifically Conjecture 1 as well as Conjecture 2 below are consistent with the family averaging assumptions made in [24, p.”
A hybrid Euler-Hadamard product for the Riemann zeta function (2007, unknown): “[3]).”
The fourth moment of Dirichlet L-functions (2006, methodology): “at a ﬁfth order pole). The smoothing function gcan be replaced with the sharp truncation t≤ T using standard techniques (however, unsurprisingly, the error term becomes larger). Recently, ﬁve authors [CFKRS] have developed a recipe for conjecturing moments of families of L-functions, including all l”
The second moment of quadratic twists of modular L-functions (2009, unknown): “Then ∑∗ 0 8d≤X (d,2)=1 L(1 2 , f ⊗ χ8d)2 ≥ (c + o(1))X log X where c = 2 π2 L(1, sym2f)3Z2(0, 0), and the value Z2(0, 0) is defined in (4.5) and (4.6).”
Moments and distribution of central L\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L$$\end{document}-values of quadr (2014, background): “s in understanding the moments of central values in families of L-functions. There are now precise, and widely believed, conjectured asymptotic formulae for moments in several important families (see [3], [9], [21], [22]), and these conjectures have been veriﬁed in a variety of cases (see for exampl”
SHARP UPPER BOUNDS FOR FRACTIONAL MOMENTS OF THE RIEMANN ZETA FUNCTION (2019, background): “1 of [3] to express that formula in terms of a multiple contour integral.”
Mod-Gaussian convergence: new limit theorems in probability and number theory (2008, unknown): “The computations above are implicit in some earlier work, for instance in the paper [4] of Conrey, Farmer, Keating, Rubinstein and Snaith on refined conjectures for moments of L-functions.”