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Multiple Dirichlet series and moments of zeta and L–functions
 PROC. OF THE GAUSSDIRICHLET CONFERENCE, GÖTTINGEN 2005, CLAY MATH. PROC., AMS
, 2001
"... This paper develops an analytic theory of Dirichlet series in several complex variables which possess sufficiently many functional equations. In the first two sections it is shown how straightforward conjectures about the meromorphic continuation and polar divisors of certain such series imply, as ..."
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Cited by 31 (8 self)
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This paper develops an analytic theory of Dirichlet series in several complex variables which possess sufficiently many functional equations. In the first two sections it is shown how straightforward conjectures about the meromorphic continuation and polar divisors of certain such series imply, as a consequence, precise asymptotics (previously conjectured via random matrix theory) for moments of zeta functions and quadratic Lseries. As an application of the theory, in a third section, we obtain the current best known error term for mean values of cubes of central values of Dirichlet Lseries. The methods utilized to derive this result are the convexity principle for functions of several complex variables combined with a knowledge of groups of functional equations for certain multiple Dirichlet series.
The cubic moment of central values of automorphic Lfunctions
 Ann. of Math
, 2000
"... 2. A review of classical modular forms 3. A review of Maass forms 4. Hecke Lfunctions ..."
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Cited by 22 (1 self)
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2. A review of classical modular forms 3. A review of Maass forms 4. Hecke Lfunctions
RankinSelberg Lfunctions in the level aspect
 Duke Math. J
"... In this paper we calculate the asymptotics of various moments of the central values of RankinSelberg convolution Lfunctions of large level, thus generalizing the results and methods of W. Duke, J. Friedlander, and H. Iwaniec and of the authors. Consequences include convexitybreaking bounds, nonva ..."
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Cited by 19 (7 self)
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In this paper we calculate the asymptotics of various moments of the central values of RankinSelberg convolution Lfunctions of large level, thus generalizing the results and methods of W. Duke, J. Friedlander, and H. Iwaniec and of the authors. Consequences include convexitybreaking bounds, nonvanishing of a positive proportion of central values, and linear independence results for certain Hecke operators. Contents
Primality testing with Gaussian periods
, 2003
"... The problem of quickly determining whether a given large integer is prime or composite has been of interest for centuries, if not longer. The past 30 years has seen a great deal of progress, leading up to the recent deterministic, polynomialtime algorithm of Agrawal, Kayal, and Saxena [2]. This new ..."
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Cited by 17 (0 self)
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The problem of quickly determining whether a given large integer is prime or composite has been of interest for centuries, if not longer. The past 30 years has seen a great deal of progress, leading up to the recent deterministic, polynomialtime algorithm of Agrawal, Kayal, and Saxena [2]. This new “AKS test ” for the primality of n involves verifying the
The distribution of the free path lengths in the periodic twodimensional Lorentz gas in the smallscatterer limit
, 2003
"... We study the free path length and the geometric free path length in the model of the periodic twodimensional Lorentz gas (Sinai billiard). We give a complete and rigorous proof for the existence of their distributions in the smallscatterer limit and explicitly compute them. As a corollary one get ..."
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Cited by 14 (6 self)
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We study the free path length and the geometric free path length in the model of the periodic twodimensional Lorentz gas (Sinai billiard). We give a complete and rigorous proof for the existence of their distributions in the smallscatterer limit and explicitly compute them. As a corollary one gets a complete proof for the existence of the constant term c = 2 − 3ln 2 + 27ζ(3) 2π2 in the asymptotic formula h(T) = −2ln ε + c + o(1) of the KS entropy of the billiard map in this model.
Quantum variance for Hecke eigenforms
 Ann. Sci. Ecole Norm. Sup
, 2004
"... ABSTRACT. – We calculate the quantum variance for the modular surface. This variance, introduced by S. Zelditch, describes the fluctuations of a quantum observable. The resulting quadratic form is then compared with the classical variance. The expectation that these two coincide only becomes true af ..."
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Cited by 9 (1 self)
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ABSTRACT. – We calculate the quantum variance for the modular surface. This variance, introduced by S. Zelditch, describes the fluctuations of a quantum observable. The resulting quadratic form is then compared with the classical variance. The expectation that these two coincide only becomes true after inserting certain subtle arithmetic factors, specifically the central values of corresponding Lfunctions. It is the offdiagonal terms in the analysis that are responsible for the rich arithmetic structure arising from the diagonalization of the quantum variance. © 2004 Elsevier SAS RÉSUMÉ. – Nous calculons la variance quantique pour la surface modulaire. Cette variance, introduite par S. Zelditch, décrit les fluctuations d’une observable quantique. La forme quadratique ainsi obtenue est comparée avec la variance classique. On s’attend à ce que toutes les deux coïncident, mais cela ne se passe qu’après inclusion de certains facteurs arithmétiques subtils, précisément les valeurs centrales des fonctions L appropriées. Les termes non diagonaux apparaissant dans l’analyse de la diagonalisation de la variance quantique sont responsables de la riche structure arithmétique. © 2004 Elsevier SAS 1.
A BURGESSLIKE SUBCONVEX BOUND FOR TWISTED LFUNCTIONS
, 2005
"... Let g be a cuspidal newform (holomorphic or Maass) of arbitrary level and nebentypus, χ a primitive character of conductor q, and s a point on the critical line ℜs = 1. It is proved that 2 L(g ⊗ χ, s) ≪ε,g,s q 1 2 − 1 8 (1−2θ)+ε, where ε> 0 is arbitrary and θ = 7 is the current known approximation ..."
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Cited by 9 (4 self)
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Let g be a cuspidal newform (holomorphic or Maass) of arbitrary level and nebentypus, χ a primitive character of conductor q, and s a point on the critical line ℜs = 1. It is proved that 2 L(g ⊗ χ, s) ≪ε,g,s q 1 2 − 1 8 (1−2θ)+ε, where ε> 0 is arbitrary and θ = 7 is the current known approximation towards the Ramanujan– 64 Petersson conjecture (which would allow θ = 0); moreover, the dependence on s and all the parameters of g is polynomial. This result is an analog of Burgess ’ classical subconvex bound for Dirichlet Lfunctions. In Appendix 2 the above result is combined with a theorem of Waldspurger and the adelic calculations of Baruch–Mao to yield an improved uniform upper bound for the Fourier coefficients of holomorphic halfintegral weight cusp forms.
The central value of the RankinSelberg Lfunctions
"... Let f be a Maass form for SL(3, Z) which is fixed and uj be an orthonormal basis of even Maass forms for SL(2, Z), we prove an asymptotic formula for the average of the product of the RankinSelberg Lfunction of f and uj and the Lfunction of uj at the central value 1/2. This implies simultaneous n ..."
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Cited by 4 (0 self)
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Let f be a Maass form for SL(3, Z) which is fixed and uj be an orthonormal basis of even Maass forms for SL(2, Z), we prove an asymptotic formula for the average of the product of the RankinSelberg Lfunction of f and uj and the Lfunction of uj at the central value 1/2. This implies simultaneous nonvanishing results of these Lfunctions at 1/2. 1