Applications of the L‐functions ratios conjectures

@article{Conrey2005ApplicationsOT,
  title={Applications of the L‐functions ratios conjectures},
  author={J. Brian Conrey and Nina C. Snaith},
  journal={Proceedings of the London Mathematical Society},
  year={2005},
  volume={94}
}
  • J. Conrey, N. Snaith
  • Published 21 September 2005
  • Mathematics
  • Proceedings of the London Mathematical Society
In upcoming papers by Conrey, Farmer and Zirnbauer there appear conjectural formulas for averages, over a family, of ratios of products of shifted L‐functions. 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 L‐functions, mollified moments of L‐functions and discrete averages over zeros of the Riemann zeta function. In particular, using the… 
View on Wiley
arxiv.org
117 Citations
Highly Influential Citations
15
Background Citations
31
Methods Citations
27
Results Citations
4

117 Citations

Triple correlation of the Riemann zeros
We use the conjecture of Conrey, Farmer and Zirn- bauer for averages of ratios of the Riemann zeta function (3) to cal- culate all the lower order terms of the triple correlation function of the…
Low-lying zeros of a family of quadratic Hecke $L$-functions via ratios conjecture
In this paper, we apply the ratio conjecture of $L$-functions to derive the lower order terms of the $1$-level density of the low-lying zeros of a family quadratic Hecke $L$-functions in the Gaussian…
A Symplectic Test of the L-Functions Ratios Conjecture
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…
Averages of ratios of the Riemann zeta-function and correlations of divisor sums
Nonlinearity has published articles containing a significant number-theoretic component since the journal was first established. We examine one thread, concerning the statistics of the zeros of the…
Unnormalized differences between zeros of L-functions
Abstract We study a subtle inequity in the distribution of unnormalized differences between imaginary parts of zeros of the Riemann zeta function, which was observed by a number of authors. We…
Low-lying zeros of elliptic curve L-functions: Beyond the Ratios Conjecture
Abstract We study the low-lying zeros of L-functions attached to quadratic twists of a given elliptic curve E defined over $\mathbb{Q}$. We are primarily interested in the family of all twists…
An orthogonal test of the L‐functions Ratios conjecture
We test the predictions of (a weakened version of) the L‐functions 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…
An orthogonal test of the $L$-functions Ratios Conjecture, II
We test the predictions of (a weakened version of) the L-functions Ratios Conjecture for the family of cuspidal newforms of weight k and level N , with either k fixed and N → ∞ through the primes or…
The Ratios Conjecture and upper bounds for negative moments of $L$-functions over function fields
We prove special cases of the Ratios Conjecture for the family of quadratic Dirichlet L–functions over function fields. More specifically, we study the average of L(1/2 + α, χD)/L(1/2 + β, χD), when…
The Ratios conjecture for real Dirichlet characters and multiple Dirichlet series
Abstract. Conrey, Farmer and Zirnbauer introduced a recipe to find asymptotic formulas for the sum of ratios of products of shifted Lfunctions. These ratios conjectures are very powerful and can be…
...
...

References

SHOWING 1-10 OF 61 REFERENCES
Low lying zeros of families of L-functions
In Iwaniec-Sarnak [IS] the percentages of nonvanishing of central values of families of GL_2 automorphic L-functions was investigated. In this paper we examine the distribution of zeros which are at…
Integral Moments of L‐Functions
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…
The non-vanishing of central values of automorphic L-functions and Landau-Siegel zeros
We describe a number of results and techniques concerning the non-vanishing of automorphic L-functions at s = ½. In particular we show that as N → ∞ at least 50% of the values L(½, f), with f varying…
Autocorrelation of ratios of L-functions
We give a new heuristic for all of the main terms in the quotient of products of L-functions averaged over a family. These conjectures generalize the recent conjectures for mean values of…
Mollification of the fourth moment of automorphic L-functions and arithmetic applications
Abstract.We compute the asymptotics of twisted fourth power moments of modular L-functions of large prime level near the critical line. This allows us to prove some new non-vanishing results on the…
Pair correlation of zeros of the zeta function.
s T— »oo and U-+Q in such a way that UL = A; here L = ̂ —logT is the average 2n density of zeros up to T and A is an arbitrary positive constant. If the same number of points were distributed at…
On the Fourth Power Moment of the Riemann Zeta-Function
Abstract The function E 2 ( R ) is used to denote the error term in the asymptotic formula for the fourth power moment of the Riemann zeta-function on the half-line. In this paper we prove several…
Averages of ratios of characteristic polynomials for the compact classical groups
Averages of ratios of characteristic polynomials for the compact classical groups are evaluated in terms of determinants whose dimensions are independent of the matrix rank. These formulas are shown…
Random matrices, Frobenius eigenvalues, and monodromy
Statements of the main results Reformulation of the main results Reduction steps in proving the main theorems Test functions Haar measure Tail estimates Large $N$ limits and Fredholm determinants…
Periodic Orbits, Spectral Statistics, and the Riemann Zeros
My purpose in this article is to review the background to some recent developments in the semiclassical theory of spectral statistics. Specifically, I will concentrate on approaches based on the…
...
...