A Symplectic Test of the L-Functions Ratios Conjecture
By Steven J. Miller. 2007
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Recently Conrey, Farmer and Zirnbauer [8, 9] conjectured formulas for the averages over a family of ratios of products of shifted L-functions. Their L-functions Ratios Conjecture predicts both the main and lower order terms for many problems, ranging from n-level correlations and densities to mollifiers and moments to vanishing at the central point. There are now many results showing agreement between the main terms of number theory and random matrix theory; however, there are very few families where the lower order terms are known. These terms often depend on subtle arithmetic properties of the family, and provide a way to break the universality of behavior. The L-functions Ratios Conjecture provides a powerful and tractable way to predict these terms. We test a specific case here, that of the 1-level density for the symplectic family of quadratic Dirichlet characters arising from even fundamental discriminants d ≤ X. For test functions supported in (−1/3, 1/3) we calculate all the lower order terms up to size O (X −½+ɛ ) and observe perfect agreement with the conjecture (for test functions supported in (−1, 1) we show agreement up to errors of size O (X −ɛ ) for any ɛ). Thus for this family and suitably restricted test functions, we completely verify the Ratios Conjecture's prediction for the 1-level density.
Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/09031a5346274a969e752bc58d50b9705317e48e
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In Algebra and Number Theory
In Geometry and Topology
In Mathematical Physics
Citations in the corpus
- Traces of high powers of the Frobenius class in the hyperelliptic ensemble (2008, background): “Corollary 3 is completely analogous to what is known in the number field setting for the corresponding case of zeta functions of quadratic fields, except for the lower order term which is different: While the coefficient of f̂(0) is as in the number field setting [14], the coefficient of f̂(1) is sp”