Results 1  10
of
30
Applications of the Lfunctions ratios conjectures
"... In upcoming papers by Conrey, Farmer and Zirnbauer there appear conjectural formulas for averages, over a family, of ratios of products of shifted Lfunctions. In this paper we will present various applications of these ratios conjectures to a wide variety of problems that are of interest in number ..."
Abstract

Cited by 22 (6 self)
 Add to MetaCart
In upcoming papers by Conrey, Farmer and Zirnbauer there appear conjectural formulas for averages, over a family, of ratios of products of shifted Lfunctions. In this paper we will present various applications of these ratios conjectures to a wide variety of problems that are of interest in number theory, such as lower order terms in the zero statistics of Lfunctions, mollified moments of Lfunctions and discrete averages over zeros of the Riemann zeta function. In particular, using the ratios conjectures we easily derive the answers to a number of notoriously difficult
Lower order terms in the 1level density for families of holomorphic cuspidal newforms
"... ABSTRACT. The KatzSarnak density conjecture states that, in the limit as the conductors tend to infinity, the behavior of normalized zeros near the central point of families of Lfunctions agree with the N → ∞ scaling limits of eigenvalues near 1 of subgroups of U(N). Evidence for this has been fo ..."
Abstract

Cited by 21 (15 self)
 Add to MetaCart
ABSTRACT. The KatzSarnak density conjecture states that, in the limit as the conductors tend to infinity, the behavior of normalized zeros near the central point of families of Lfunctions agree with the N → ∞ scaling limits of eigenvalues near 1 of subgroups of U(N). Evidence for this has been found for many families by studying the nlevel densities; for suitably restricted test functions the main terms agree with random matrix theory. In particular, all oneparameter families of elliptic curves with rank r over Q(T) and the same distribution of signs of functional equations have the same limiting behavior. We break this universality and find family dependent lower order correction terms in many cases; these lower order terms have applications ranging from excess rank to modeling the behavior of zeros near the central point, and depend on the arithmetic of the family. We derive an alternate form of the explicit formula for GL(2) Lfunctions which simplifies comparisons, replacing sums over powers of Satake parameters by sums of the moments of the Fourier coefficients λf(p). Our formula highlights the differences that we expect to exist from families whose Fourier coefficients obey different laws (for example, we expect SatoTate to hold only for nonCM families of elliptic curves). Further, by the work of Rosen and Silverman we expect lower order biases to the Fourier coefficients in families of elliptic curves with rank over Q(T); these biases can be seen in our expansions. We analyze several families of elliptic curves and see different lower order corrections, depending on whether or not the family has complex multiplication, a forced torsion point, or nonzero rank over Q(T). 1.
Autocorrelation of ratios of Lfunctions
 COMM. NUMBER THEORY AND PHYSICS
, 2007
"... We give a new heuristic for all of the main terms in the quotient of products of Lfunctions averaged over a family. These conjectures generalize the recent conjectures for mean values of Lfunctions. Comparison is made to the analogous quantities for the characteristic polynomials of matrices ave ..."
Abstract

Cited by 21 (3 self)
 Add to MetaCart
(Show Context)
We give a new heuristic for all of the main terms in the quotient of products of Lfunctions averaged over a family. These conjectures generalize the recent conjectures for mean values of Lfunctions. Comparison is made to the analogous quantities for the characteristic polynomials of matrices averaged over a classical compact group.
A symplectic test of the Lfunctions ratios conjecture
 Int. Math. Res. Notices, 2008, article ID rnm
"... ABSTRACT. Recently Conrey, Farmer and Zirnbauer [CFZ1, CFZ2] conjectured formulas for the averages over a family of ratios of products of shifted Lfunctions. Their Lfunctions Ratios Conjecture predicts both the main and lower order terms for many problems, ranging from nlevel correlations and den ..."
Abstract

Cited by 16 (6 self)
 Add to MetaCart
ABSTRACT. Recently Conrey, Farmer and Zirnbauer [CFZ1, CFZ2] conjectured formulas for the averages over a family of ratios of products of shifted Lfunctions. Their Lfunctions Ratios Conjecture predicts both the main and lower order terms for many problems, ranging from nlevel 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 Lfunctions Ratios Conjecture provides a powerful and tractable way to predict these terms. We test a specific case here, that of the 1level 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 −1/2+ǫ) and observe perfect agreement with the conjecture (for test functions supported in (−1, 1) we show agreement up to errors of size O(X −ǫ) for any ǫ). Thus for this family and suitably restricted test functions, we completely verify the Ratios Conjecture’s prediction for the 1level density. 1.
An orthogonal test of the LFunctions Ratios Conjecture
"... ABSTRACT. We test the predictions of the Lfunctions Ratios Conjecture for the family of cuspidal newforms of weight k and level N, with either k fixed and N → ∞ through the primes or N = 1 and k → ∞. We study the main and lower order terms in the 1level density. We provide evidence for the Ratios ..."
Abstract

Cited by 9 (6 self)
 Add to MetaCart
ABSTRACT. We test the predictions of the Lfunctions Ratios Conjecture for the family of cuspidal newforms of weight k and level N, with either k fixed and N → ∞ through the primes or N = 1 and k → ∞. We study the main and lower order terms in the 1level density. We provide evidence for the Ratios Conjecture by computing and confirming its predictions up to a power savings in the family’s cardinality, at least for test functions whose Fourier transforms are supported in (−2, 2). We do this both for the weighted and unweighted 1level density (where in the weighted case we use the Petersson weights), thus showing that either formulation may be used. These two 1level densities differ by a term of size 1 / log(k 2 N). Finally, we show that there is another way of extending the sums arising in the Ratios Conjecture, leading to a different answer (although the answer is such a lower order term that it is hopeless to observe which is correct). 1.
Surpassing the Ratios Conjecture in the 1level density of Dirichlet Lfunctions
 ALGEBRA & NUMBER THEORY
, 2012
"... ..."
Triple correlation of the Riemann zeros
"... We use the conjecture of Conrey, Farmer and Zirnbauer for averages of ratios of the Riemann zeta function [11] to calculate all the lower order terms of the triple correlation function of the Riemann zeros. A previous approach was suggested by Bogomolny and Keating [6] taking inspiration from semi ..."
Abstract

Cited by 7 (2 self)
 Add to MetaCart
(Show Context)
We use the conjecture of Conrey, Farmer and Zirnbauer for averages of ratios of the Riemann zeta function [11] to calculate all the lower order terms of the triple correlation function of the Riemann zeros. A previous approach was suggested by Bogomolny and Keating [6] taking inspiration from semiclassical methods. At that point they did not write out the answer explicitly, so we do that here, illustrating that by our method all the lower order terms down to the constant can be calculated rigourously if one assumes the ratios conjecture of Conrey, Farmer and Zirnbauer. Bogomolny and Keating [4] returned to their previous results simultaneously with this current work, and have written out the full expression. The result presented in this paper agrees precisely with their formula, as well as with our numerical computations, which we include here. We also include an alternate proof of the triple correlation of eigenvalues from random U(N) matrices which follows a nearly identical method to that for the Riemann zeros, but is based on
On absolute moments of characteristic polynomials of a certain class of complex random matrices
, 2006
"... ..."
Riemann zeros and random matrix theory
, 2009
"... In the past dozen years random matrix theory has become a useful tool for conjecturing answers to old and important questions in number theory. It was through the Riemann zeta function that the connection with random matrix theory was first made in the 1970s, and although there has also been much re ..."
Abstract

Cited by 4 (0 self)
 Add to MetaCart
(Show Context)
In the past dozen years random matrix theory has become a useful tool for conjecturing answers to old and important questions in number theory. It was through the Riemann zeta function that the connection with random matrix theory was first made in the 1970s, and although there has also been much recent work concerning other varieties of Lfunctions, this article will concentrate on the zeta function as the simplest example illustrating the role of random matrix theory. 1