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29
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 ..."
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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.
The fourth moment of Dirichlet Lfunctions
 Ann. of Math
"... Abstract We compute the fourth moment of Dirichlet Lfunctions at the central point for prime moduli, with a power savings in the error term. ..."
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Cited by 12 (1 self)
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Abstract We compute the fourth moment of Dirichlet Lfunctions at the central point for prime moduli, with a power savings in the error term.
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 ..."
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Cited by 7 (2 self)
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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
Mean values with cubic characters
, 2009
"... We investigate various mean value problems involving order three primitive Dirichlet characters. In particular, we obtain an asymptotic formula for the first moment of central values of the Dirichlet Lfunctions associated to this family, with a power savings in the error term. We also obtain a lar ..."
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Cited by 7 (1 self)
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We investigate various mean value problems involving order three primitive Dirichlet characters. In particular, we obtain an asymptotic formula for the first moment of central values of the Dirichlet Lfunctions associated to this family, with a power savings in the error term. We also obtain a largesieve type result for order three (and six) Dirichlet characters.
THE SIXTH POWER MOMENT OF DIRICHLET LFUNCTIONS
, 710
"... Understanding moments of families of Lfunctions has long been an important subject with many number theoretic applications. Quite often, the application is to bound the error term in an asymptotic expression of the average of an arithmetic function. Also, a good bound for a moments can be used to o ..."
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Cited by 6 (2 self)
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Understanding moments of families of Lfunctions has long been an important subject with many number theoretic applications. Quite often, the application is to bound the error term in an asymptotic expression of the average of an arithmetic function. Also, a good bound for a moments can be used to obtain a pointwise bound for an individual Lfunction in the
A RANDOM MATRIX MODEL FOR ELLIPTIC CURVE LFUNCTIONS OF FINITE CONDUCTOR
, 2011
"... Abstract. We propose a random matrix model for families of elliptic curve Lfunctions of finite conductor. A repulsion of the critical zeros of these Lfunctions away from the center of the critical strip was observed numerically by S. J. Miller in 2006 [50]; such behaviour deviates qualitatively fr ..."
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Cited by 5 (3 self)
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Abstract. We propose a random matrix model for families of elliptic curve Lfunctions of finite conductor. A repulsion of the critical zeros of these Lfunctions away from the center of the critical strip was observed numerically by S. J. Miller in 2006 [50]; such behaviour deviates qualitatively from the conjectural limiting distribution of the zeros (for large conductors this distribution is expected to approach the onelevel density of eigenvalues of orthogonal matrices after appropriate rescaling). Our purpose here is to provide a random matrix model for Miller’s surprising discovery. We consider the family of even quadratic twists of a given elliptic curve. The main ingredient in our model is a calculation of the eigenvalue distribution of random orthogonal matrices whose characteristic polynomials are larger than some given value at the symmetry point in the spectra. We call this subensemble of SO(2N) the excised orthogonal ensemble. The sievingoff of matrices with small values of the characteristic polynomial is akin to the discretization of the central values of Lfunctions implied by the formulæ of Waldspurger and KohnenZagier. The cutoff scale
Regularization for zeta functions with physical applications I,
, 609
"... Abstract We have proposed a regularization technique and applied it to the Eu ler product of the zeta functions in the part one. In this paper that is the second part of the trilogy, we aim the nature of the nontrivial zero for the Riemann zeta function which gives us another evidence to demonstra ..."
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Abstract We have proposed a regularization technique and applied it to the Eu ler product of the zeta functions in the part one. In this paper that is the second part of the trilogy, we aim the nature of the nontrivial zero for the Riemann zeta function which gives us another evidence to demonstrate the Riemann hypotheses by way of the approximate functional equation.Some other results on the critical line are p resented using the relations between the Euler product and the deformed summation representations in the critical strip. We also discuss a set of equations which yields the primes and the zeros of the zeta functions. In part three, we will focus on physical applicat ions using these outcomes.
Special values of symmetric power Lfunctions and Hecke Eigenvalues
 JOURNAL DE THÉORIE DES NOMBRES DE BORDEAUX 19 (2007), 703–753
, 2007
"... We compute the moments of Lfunctions of symmetric powers of modular forms at the edge of the critical strip, twisted by the central value of the Lfunctions of modular forms. We show ..."
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Cited by 4 (2 self)
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We compute the moments of Lfunctions of symmetric powers of modular forms at the edge of the critical strip, twisted by the central value of the Lfunctions of modular forms. We show
Symmetry Transitions in Random Matrix Theory & Lfunctions
, 2008
"... We compute the moments of the characteristic polynomials of random orthogonal and symplectic matrices, defined by averages with respect to Haar measure on SO(2N) and U Sp(2N), to leading order as N → ∞, on the unit circle as functions of the angle θ measured from one of the two symmetry points in t ..."
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Cited by 4 (1 self)
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We compute the moments of the characteristic polynomials of random orthogonal and symplectic matrices, defined by averages with respect to Haar measure on SO(2N) and U Sp(2N), to leading order as N → ∞, on the unit circle as functions of the angle θ measured from one of the two symmetry points in the eigenvalue spectrum {exp(±iθn)}1≤n≤N. Our results extend previous formulae that relate just to the symmetry points, i.e. to θ = 0. Local spectral statistics are expected to converge to those of random unitary matrices in the limit as N → ∞ when θ is fixed, and to show a transition from the orthogonal or symplectic to the unitary forms on the scale of the mean eigenvalue spacing: if θ = πy/N they become functions of y in the limit when N → ∞. We verify that this is true for the spectral twopoint correlation function, but show that it is not true for the moments of the characteristic polynomials, for which the leading order asymptotic approximation is a function of θ rather than y. Symmetry points therefore influence the moments asymptotically far away on the scale of the mean eigenvalue spacing. We also investigate the moments of the logarithms of the characteristic polynomials in the same context. The moments of the characteristic polynomials of
Pair correlation of the zeros of the derivative of the Riemann ξfunction
, 2008
"... The complex zeros of the Riemannn zetafunction are identical to the zeros of the Riemann xifunction, ξ(s). Thus, if the Riemann Hypothesis is true for the zetafunction, it is true for ξ(s). Since ξ(s) is entire, the zeros of ξ ′ (s), its derivative, would then also satisfy a Riemann Hypothesis. ..."
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Cited by 3 (1 self)
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The complex zeros of the Riemannn zetafunction are identical to the zeros of the Riemann xifunction, ξ(s). Thus, if the Riemann Hypothesis is true for the zetafunction, it is true for ξ(s). Since ξ(s) is entire, the zeros of ξ ′ (s), its derivative, would then also satisfy a Riemann Hypothesis. We investigate the pair correlation function of the zeros of ξ′(s) under the assumption that the Riemann Hypothesis is true. We then deduce consequences about the size of gaps between these zeros and the proportion of these zeros that are simple.