Random Matrix Theory and L-Functions at s= 1/2

By Jon P Keating, N. Snaith. 2000

375 citations recorded by the source corpus.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/7d30407b804e487ff919cba38cfc1591d04503c4

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In Algebra and Number Theory

In Discrete Mathematics and Combinatorics

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Citations in the corpus

Universality for mathematical and physical systems (2006, background): “elationship between random polynomials whose roots are given by the eigenvalues of a matrix from some random ensemble, and the moments of the Lfunctions on the critical line Rez = 1 2 (see [KeaSna1], [KeaSna2]). The discovery of Montgomery/Odlyzko counts as one of the major developments in analytic ”
Sharp conditional bounds for moments of the Riemann zeta function (2013, methodology): “Then for all large X we have (cid:0) (cid:1) Keating and Snaith [9] conjectured an asymptotic formula for the left hand side in Theorem 2, building on work of Conrey and Farmer [2], and the upper bound in Theorem 2 matches that conjecture.”
How to generate random matrices from the classical compact groups (2006, background): “Indeed, the work of Montgomery [11], Odlyzko [12], Katz and Sarnak [6], Keating and Snaith [7, 8] and Rubinstein [13] has shown beyond doubt that the local statistical properties of the”
Lower bounds for moments of L-functions: symplectic and orthogonal examples (2006, background): “[7] J. P. Keating & N. C. Snaith, Random matrix theory and L-functions at s = 1/2, Comm. in Math.”
Conjectures for the integral moments and ratios of L-functions over function fields (2014, background): “0) × (−1)k(k−1)/2 (2πi)k 1 k! I ··· I e Pk j=1vj Y 1≤i<j≤k 1 vi +vj ∆(v2 1,...,v 2 k) 2 Qk j=1 v 2k j dv1 ...dvk. Using equation (3.36) from [CFKRS2], Lemma 2.5.2 from [CFKRS], and the result from [KeS2] for the moments at the symmetry point for the symplectic ensemble completes the proof of the ”
Applications of the L‐functions ratios conjectures (2005, background): “d over a family. These are global, rather than local, statistics. A characteristic feature of a global statistic is that an arithmetic factor appears in the leading order term. In the original papers [29,30] the leading term was a product of the corresponding moment of a characteristic polynomial fr”
Autocorrelation of ratios of L-functions (2007, background): “Conjectures for the moments of L-functions have been sought for many decades, with very little progress until the introduction of random matrix theory into the subject [3, 26, 27 ].”
A Symplectic Test of the L-Functions Ratios Conjecture (2007, background): “One example of this is that zeros at height T are modeled not by the N → ∞ scaling limits of a classical compact group but by matrices of size N ∼ log(T/2π) [KeSn1, KeSn2 ].”
Moments of the Riemann zeta function (2006, background): “ositive real numbers k. See also the elegant note [1] giving such a bound assuming RH, and [20] for the best known constants implicit in these lower bounds. Analogous conjectures exist (see [2], [4], [11]) for moments of central values of L-functions in families, and in many cases lower bounds of th”