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573
CYCLE INTEGRALS OF THE JFUNCTION AND MOCK MODULAR FORMS
"... In this paper we construct certain mock modular forms of weight 1/2 whose Fourier coefficients are cycle integrals of the modular jfunction and whose shadows are weakly holomorphic forms of weight 3/2. As an application we construct through a Shimuratype lift a holomorphic function that transforms ..."
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Cited by 23 (5 self)
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In this paper we construct certain mock modular forms of weight 1/2 whose Fourier coefficients are cycle integrals of the modular jfunction and whose shadows are weakly holomorphic forms of weight 3/2. As an application we construct through a Shimuratype lift a holomorphic function that transforms with a rational period function having poles at certain real quadratic integers. This function yields a real quadratic analogue of the Borcherds product.
A HYBRID EULERHADAMARD PRODUCT FOR THE RIEMANN ZETA FUNCTION
"... We use a smoothed version of the explicit formula to find an accurate pointwise approximation to the Riemann zeta function as a product over its nontrivial zeros multiplied by a product over the primes. We model the first product by characteristic polynomials of random matrices. This provides a stat ..."
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Cited by 21 (9 self)
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We use a smoothed version of the explicit formula to find an accurate pointwise approximation to the Riemann zeta function as a product over its nontrivial zeros multiplied by a product over the primes. We model the first product by characteristic polynomials of random matrices. This provides a statistical model of the zeta function which involves the primes in a natural way. We then employ the model in a heuristic calculation of the moments of the modulus of the zeta function on the critical line. For the second and fourth moments, we establish all of the steps in our approach rigorously. This calculation illuminates recent conjectures for these moments based
Statistics for lowlying zeros of symmetric power Lfunctions in the level aspect
, 2007
"... ABSTRACT. We study onelevel and twolevel densities for low lying zeros of symmetric power Lfunctions in the level aspect. It allows us to completely determine the symmetry types of some families of symmetric power Lfunctions with prescribed sign of functional equation. We also compute themoments ..."
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Cited by 21 (1 self)
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ABSTRACT. We study onelevel and twolevel densities for low lying zeros of symmetric power Lfunctions in the level aspect. It allows us to completely determine the symmetry types of some families of symmetric power Lfunctions with prescribed sign of functional equation. We also compute themoments of onelevel density and exhibit mockGaussian behavior discovered by Hughes & Rudnick. CONTENTS
Sumproduct Estimates in Finite Fields via Kloosterman Sums
"... We establish improved sumproduct bounds in finite fields using incidence theorems based on bounds for classical Kloosterman and related sums. ..."
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Cited by 21 (4 self)
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We establish improved sumproduct bounds in finite fields using incidence theorems based on bounds for classical Kloosterman and related sums.
More than 41% of the zeros of the zeta function are on the critical line
 Acta Arith
"... The location of the zeros of the Riemann zeta function is one of the most fascinating subjects in number theory. In this paper we study the percent of zeros lying on the critical line. With the use of a new twopiece mollifier, we make a modest improvement on this important problem. ..."
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Cited by 20 (7 self)
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The location of the zeros of the Riemann zeta function is one of the most fascinating subjects in number theory. In this paper we study the percent of zeros lying on the critical line. With the use of a new twopiece mollifier, we make a modest improvement on this important problem.
Average Frobenius distribution of elliptic curves
, 2005
"... The SatoTate conjecture asserts that given an elliptic curve without complex multiplication, the primes whose Frobenius elements have their trace in a given interval (2α √ p, 2β √ p) 1 − t2 dt. We prove that this conjecture is true on average in a have density given by 2 π more general setting. ..."
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Cited by 20 (5 self)
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The SatoTate conjecture asserts that given an elliptic curve without complex multiplication, the primes whose Frobenius elements have their trace in a given interval (2α √ p, 2β √ p) 1 − t2 dt. We prove that this conjecture is true on average in a have density given by 2 π more general setting.
Lowerorder terms of the 1level density of families of elliptic curves
, 2008
"... The KatzSarnak philosophy predicts that statistics of zeros of families of Lfunctions are strikingly universal. However, subtle arithmetical differences between families of the same symmetry type can be detected by calculating lowerorder terms of the statistics of interest. In this paper we calcu ..."
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Cited by 19 (2 self)
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The KatzSarnak philosophy predicts that statistics of zeros of families of Lfunctions are strikingly universal. However, subtle arithmetical differences between families of the same symmetry type can be detected by calculating lowerorder terms of the statistics of interest. In this paper we calculate lowerorder terms of the 1level density of some families of elliptic curves. We show that there are essentially two different effects on the distribution of lowlying zeros. First, lowlying zeros are more numerous in families of elliptic curves E with relatively large numbers of points (mod p). Second, and somewhat surprisingly, a family with a relatively large number of primes of bad reduction has relatively fewer lowlying zeros (essentially because if the elliptic curve E has bad reduction at p then the local factor Lp(1/2,E) is larger than otherwise expected). We also show that the lower order term can grow arbitrarily large by taking a biased family with a relatively large number of points (mod p) for all small primes p.
Hybrid bounds for twisted Lfunctions
 MR 2431250 (2009e:11094
"... Abstract. The aim of this paper is to derive bounds on the critical line <s = 1 2 for Lfunctions attached to twists f ⊗ χ of a primitive cusp form f of level N and a primitive character modulo q that break convexity simultaneously in the s and q aspects. If f has trivial nebentypus, it is shown ..."
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Cited by 19 (4 self)
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Abstract. The aim of this paper is to derive bounds on the critical line <s = 1 2 for Lfunctions attached to twists f ⊗ χ of a primitive cusp form f of level N and a primitive character modulo q that break convexity simultaneously in the s and q aspects. If f has trivial nebentypus, it is shown that L(f ⊗ χ, s) (N sq)εN 45 (sq) 12 − 140, where the implied constant depends only on ε> 0 and the archimedean parameter of f. To this end, two independent methods are employed to show
Reflection Principles and Bounds for Class Group Torsion
, 2007
"... We introduce a new method to bound ℓtorsion in class groups, combining analytic ideas with reflection principles. This gives, in particular, new bounds for the 3torsion part of class groups in quadratic, cubic and quartic number fields, as well as bounds for certain families of higher degree field ..."
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Cited by 19 (1 self)
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We introduce a new method to bound ℓtorsion in class groups, combining analytic ideas with reflection principles. This gives, in particular, new bounds for the 3torsion part of class groups in quadratic, cubic and quartic number fields, as well as bounds for certain families of higher degree fields and for higher ℓ. Conditionally on GRH, we obtain a nontrivial bound for ℓtorsion in the class group of a general number field.