[Show abstract][Hide abstract] ABSTRACT: In this paper we examine n-correlation for either the eigenvalues of a unitary group of random matrices or for the zeros of a unitary family of L-functions in the important situation when the correlations are detected via test functions whose Fourier transforms have limited support. This problem first came to light in the work of Rudnick and Sarnak in their study of the n-correlation of zeros of a fairly general automorphic L-function. They solved the simplest instance of this problem when the total support was most severely limited, but had to work extremely hard to show their result matched random matrix theory in the appropriate limit. This is because they were comparing their result to the familiar determinantal expressions for n-correlation that arise naturally in random matrix theory. In this paper we deal with arbitrary support and show that there is another expression for the n-correlation of eigenvalues that translates easily into the number theory case and allows for immediate identification of which terms survive the restrictions placed on the support of the test function.
[Show abstract][Hide abstract] ABSTRACT: We introduce the cotangent sum c(h/k)=-∑ h=1 k-1 a kcotπah k and prove that it satisfies hc(h/k)+c(k/h)−1 πk=g(h/k), where g is a function which is analytic on the complex plane minus the negative real axis. The sum arises in connection with the Nyman-Beurling approach to the Riemann hypothesis.
International Mathematics Research Notices 01/2013; 2013(24). DOI:10.1093/imrn/rns211 · 1.10 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: In this paper we examine $n$-correlation for either the eigenvalues of a
unitary group of random matrices or for the zeros of a unitary family of
$L$-functions in the important situation when the correlations are detected via
test functions whose Fourier transforms have limited support. This problem
first came to light in the work of Rudnick and Sarnak in their study of the
$n$-correlation of zeros of a fairly general automorphic $L$-function. They
solved the simplest instance of this problem when the total support was most
severely limited, but had to work extremely hard to show their result matched
random matrix theory in the appropriate limit. This is because they were
comparing their result to the familiar determinantal expressions for
$n$-correlation that arise naturally in random matrix theory. In this paper we
deal with arbitrary support and show that there is another expression for the
$n$-correlation of eigenvalues that translates easily into the number theory
case and allows for immediate identification of which terms survive the
restrictions placed on the support of the test function.
[Show abstract][Hide abstract] ABSTRACT: The family of symmetric powers of an $L$-function associated with an elliptic
curve with complex multiplication has received much attention from algebraic,
automorphic and p-adic points of view. Here we examine one explicit such family
from the perspectives of classical analytic number theory and random matrix
theory, especially focusing on evidence for the symmetry type of the family. In
particular, we investigate the values at the central point and give evidence
that this family can be modeled by ensembles of orthogonal matrices. We prove
an asymptotic formula with power savings for the average of these L-values,
which reproduces, by a completely different method, an asymptotic formula
proven by Greenberg and Villegas--Zagier. We give an upper bound for the second
moment which is conjecturally too large by just one logarithm. We also give an
explicit conjecture for the second moment of this family, with power savings.
Finally, we compute the one level density for this family with a test function
whose Fourier transform has limited support. It is known by the work of
Villegas -- Zagier that the subset of these L-functions from our family which
have even functional equations never vanish; we show to what extent this result
is reflected by our analytic results.
[Show abstract][Hide abstract] ABSTRACT: Stephen D. Miller showed that, assuming the generalized Riemann Hypothesis,
every entire $L$-function of real archimedian type has a zero in the interval
$\frac12+i t$ with $-t_0 < t < t_0$, where $t_0\approx 14.13$ corresponds to
the first zero of the Riemann zeta function. We give an example of a self-dual
degree-4 $L$-function whose first positive imaginary zero is at $t_1\approx
14.496$. In particular, Miller's result does not hold for general
$L$-functions. We show that all $L$-functions satisfying some additional
(conjecturally true) conditions have a zero in the interval $(-t_2,t_2)$ with
$t_2\approx 22.661$.
Journal of Number Theory 11/2012; 147. DOI:10.1016/j.jnt.2014.07.023 · 0.59 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: The Nyman-Beurling-Báez-Duarte approach to the Riemann hypothesis asserts that the Riemann hypothesis is true if and only if lim N→∞ d N 2 =0, where d N 2 =inf A N 1 2π∫ -∞ ∞ 1-ζA N 1 2-it 2 dt 1/4+t 2 , and the infimum is over all Dirichlet polynomials A N (s)=∑ n=1 N a n /n s of length N. An open question is to determine what the rate of convergence of dn to zero is, assuming the Riemann hypothesis. L. Baez-Duarte et al. Adv. Math. 149, No. 1, 130–144, (2000; Zbl 1008.11032); Ramanujan J. 9, No. 1–2, 215–240 (2005; Zbl 1173.11343)] showed (unconditionally) that d N 2 can not decay faster than a constant times 1/logN. More precisely, they showed that lim N→∞ infd N 2 logN≥∑ ℜ(ρ)=1/2 1 |ρ| 2 , where the sum is restricted to distinct zeros of the Riemann zeta function on the critical line. Actually one expects that d N 2 ∼1 logN∑ ℜ(ρ)=1/2 m(ρ) 2 |ρ| 2 ,(1) where m(ρ) denotes the multiplicity of ρ. It is the purpose of this note to prove (1) under the Riemann Hypothesis and assuming a mild condition on the growth of the mean value of 1/|ζ ' (ρ)| 2 over the non-trivial zeros |ρ|≤T of ζ(s).
Proceedings of the Steklov Institute of Mathematics 11/2012; 280(2). DOI:10.1134/S0081543813030036 · 0.30 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: We use the asymptotic large sieve, developed by the authors, to prove that if
the Generalized Riemann Hypothesis is true, then there exist many Dirichlet
L-functions that have a pair of consecutive zeros closer together than 0.37
times their average spacing. More generally, we investigate zero spacings
within the family of twists by Dirichlet characters of a fixed L-function and
give precise bounds for small gaps which depend only on the degree of the
L-function.
[Show abstract][Hide abstract] ABSTRACT: We show that all but 5 of the zeros of the period polynomial associated to a
Hecke cusp form are on the unit circle.
International Mathematics Research Notices 01/2012; 2013(20). DOI:10.1093/imrn/rns183 · 1.10 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: We show that if a real trigonometric polynomial has few real roots, then the trigonometric polynomial obtained by writing the coefficients in reverse order must have many real roots. This is used to show that a class of random trigonometric polynomials has, on average, many real roots. In the case that the coefficients of a real trigonometric polynomial are independently and identically distributed, but with no other assumptions on the distribution, the expected fraction of real zeros is at least one-half. This result is best possible.
Proceedings of the American Mathematical Society 01/2009; DOI:10.1090/S0002-9939-08-09776-1 · 0.68 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: We present a new approach to obtaining the lower order terms for $n$-correlation of the zeros of the Riemann zeta function. Our approach is based on the `ratios conjecture' of Conrey, Farmer, and Zirnbauer. Assuming the ratios conjecture we prove a formula which explicitly gives all of the lower order terms in any order correlation. Our method works equally well for random matrix theory and gives a new expression, which is structurally the same as that for the zeta function, for the $n$-correlation of eigenvalues of matrices from U(N).
Communications in Number Theory and Physics 04/2008; DOI:10.4310/CNTP.2008.v2.n3.a1 · 1.43 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: We prove a formula, with power savings, for the sixth moment of Dirichlet
L-functions averaged over moduli $q$, over primitive characters $\chi$ modulo
$q$, and over the critical line. Our formula agrees precisely with predictions
motivated by random matrix theory. In particular, the constant 42 appears as a
factor in the leading order term, exactly as is predicted for the sixth moment
of the Riemann zeta-function.
[Show abstract][Hide abstract] ABSTRACT: We verify the conjecture of [CFKRS] for the mean square near the critical point of Dirichlet L-functions for a composite modulus q. We also prove a kind of reciprocity formula when the second moment for a prime modulus is twisted by a character evaluated at a different prime.
[Show abstract][Hide abstract] ABSTRACT: We prove that a Dirichlet series with a functional equation and Euler product of a particular form can only arise from a holomorphic cusp form on the Hecke congruence group Γ0(13). The proof does not assume a functional equation for the twists of the Dirichlet series. The main new ingredient is a generalization of the familiar Weil's lemma that played a prominent role in previous converse theorems.
Journal of Number Theory 02/2007; 122(2-122):314-323. DOI:10.1016/j.jnt.2006.05.017 · 0.59 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: We describe an algorithm for obtaining explicit expressions for lower terms for the conjectured full asymptotics of the moments of the Riemann zeta function, and give two distinct methods for obtaining numerical values of these coefficients. We also provide some numerical evidence in favor of the conjecture.
Journal of Number Theory 01/2007; 128(6-128):1516-1554. DOI:10.1016/j.jnt.2007.05.013 · 0.59 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: We investigate the moments of the derivative, on the unit circle, of characteristic polynomials of random unitary matrices and use this to formulate a conjecture for the moments of the derivative of the Riemann ζ function on the critical line. We do the same for the analogue of Hardy’s Z-function, the characteristic polynomial multiplied by a suitable factor to make it real on the unit circle. Our formulae are expressed in terms of a determinant of a matrix whose entries involve the I-Bessel function and, alternately, by a combinatorial sum.
[Show abstract][Hide abstract] ABSTRACT: We use the conjecture of Conrey, Farmer and Zirnbauer for averages of ratios
of the Riemann zeta function to calculate all the lower order terms of the
triple correlation function of the Riemann zeros. A previous approach was
suggested in 1996 by Bogomolny and Keating taking inspiration from
semi-classical 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
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 the theorem for averages of ratios of
characteristic polynomials.
Journal de Theorie des Nombres de Bordeaux 10/2006; DOI:10.5802/jtnb.616 · 0.41 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: We prove that a Dirichlet series with a functional equation and Euler product of a particular form can only arise from a holomorphic cusp form on the Hecke congruence group $\Gamma_0(13)$. The proof does not assume a functional equation for the twists of the Dirichlet series. The main new ingredient is a generalization of the familiar Weil's lemma that played a prominent role in previous converse theorems.
[Show abstract][Hide abstract] ABSTRACT: Within the study of arithmetical Dirichlet series, those that have a functional equation and Euler product are of particular interest. In 1989 Selberg described a class $\mathcal{S}$ of Dirichlet series through a set of four axioms which possibly contain all of these interesting Dirichlet series and made a number of interesting conjectures. In particular, he conjectured the Riemann Hypothesis for this class. We prove that one consequence of the Riemann Hypothesis for functions in $\mathcal{S}$ is the generalized Lindelöf Hypothesis. Moreover, we give an example of a function $D$ which satisfies the first three of Selberg's axioms but fails the Lindelöf Hypothesis in the $Q$ aspect.
Functiones et Approximatio 01/2006; 36(1). DOI:10.7169/facm/1229616442
[Show abstract][Hide abstract] ABSTRACT: Conjectured links between the distribution of values taken by the characteristic polynomials of random orthogonal matrices and that for certain families of $L$-functions at the center of the critical strip are used to motivate a series of conjectures concerning the value distribution of the Fourier coefficients of half-integral-weight modular forms related to these $L$-functions. Our conjectures may be viewed as being analogous to the Sato-Tate conjecture for integral-weight modular forms. Numerical evidence is presented in support of them.