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24
Mean values of Lfunctions and symmetry
 Int. Math. Res. Notices
"... Abstract. Recently Katz and Sarnak introduced the idea of a symmetry group attached to a family of L–functions, and they gave strong evidence that the symmetry group governs many properties of the distribution of zeros of the L–functions. We consider the mean–values of the L–functions and the mollif ..."
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Cited by 54 (13 self)
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Abstract. Recently Katz and Sarnak introduced the idea of a symmetry group attached to a family of L–functions, and they gave strong evidence that the symmetry group governs many properties of the distribution of zeros of the L–functions. We consider the mean–values of the L–functions and the mollified mean–square of the L–functions and find evidence that these are also governed by the symmetry group. We use recent work of Keating and Snaith to give a complete description of these mean values. We find a connection to the Barnes–Vignéras Γ2–function and to a family of self–similar functions. 1.
Linear statistics of lowlying zeros of L–functions”, (preprint
"... Abstract. We consider linear statistics of the scaled zeros of Dirichlet L– functions, and show that the first few moments converge to the Gaussian moments. The number of Gaussian moments depends on the particular statistic considered. The same phenomenon is found in Random Matrix Theory, where we c ..."
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Cited by 50 (5 self)
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Abstract. We consider linear statistics of the scaled zeros of Dirichlet L– functions, and show that the first few moments converge to the Gaussian moments. The number of Gaussian moments depends on the particular statistic considered. The same phenomenon is found in Random Matrix Theory, where we consider linear statistics of scaled eigenphases for matrices in the unitary group. In that case the higher moments are no longer Gaussian. We conjecture that this also happens for Dirichlet L–functions. 1.
High moments of the Riemann zetafunction
 Duke Math. J
"... In 1918 G. Hardy and J. Littlewood proved an asymptotic estimate for the Second moment of the modulus of the Riemann zetafunction on the segment [1/2,1/2+iT] in the complex plane, as T tends to infinity. In 1926 Ingham proved an asymptotic estimate for the fourth moment. However, since Ingham’s res ..."
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Cited by 44 (4 self)
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In 1918 G. Hardy and J. Littlewood proved an asymptotic estimate for the Second moment of the modulus of the Riemann zetafunction on the segment [1/2,1/2+iT] in the complex plane, as T tends to infinity. In 1926 Ingham proved an asymptotic estimate for the fourth moment. However, since Ingham’s result, nobody has proved an asymptotic formula for any higher moment. Recently J. Conrey and A. Ghosh conjectured a formula for the sixth moment. We develop a new heuristic method to conjecture the asymptotic size of both the sixth and eighth moments. Our estimate for the sixth moment agrees with and strongly supports, in a sense made clear in the paper, the one conjectured by Conrey and Ghosh. Moreover, both our sixth and eighth moment estimates agree with those conjectured recently by J. Keating and N. Snaith based on modeling the zetafunction by characteristic polynomials of random matrices from the Gaussian unitary ensemble. Our method uses a conjectural form of the approximate functional equation for the zetafunction, a conjecture on the behavior of additive divisor sums, and D. Goldston and S. Gonek’s mean value theorem for
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 27 (1 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
ANALYTIC PROBLEMS FOR ELLIPTIC CURVES
, 2005
"... Abstract. We consider some problems of analytic number theory for elliptic curves which can be considered as analogues of classical questions around the distribution of primes in arithmetic progressions to large moduli, and to the question of twin primes. This leads to some local results on the dist ..."
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Cited by 15 (0 self)
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Abstract. We consider some problems of analytic number theory for elliptic curves which can be considered as analogues of classical questions around the distribution of primes in arithmetic progressions to large moduli, and to the question of twin primes. This leads to some local results on the distribution of the group structures of elliptic curves defined over a prime finite field, exhibiting an interesting dichotomy for the occurence of the possible groups. (This paper was initially written in 2000/01, but after a four year wait for a referee report, it is now withdrawn and deposited in the arXiv). Contents
Reduction mod p of subgroups of the MordellWeil group of an elliptic curve
 INT J. OF NUMBER THEORY
"... Let E be an elliptic curve defined over Q. Let Γ be a free subgroup of rank r of E(Q). For any prime p of good reduction, let Γp be the reduction of Γ modulo p and Ep be the reduction of E modulo p. We prove that if E has CM then for all but o(x / log x) of primes p ≤ x, Γp  ≥ p r r+2 +ɛ(p), wher ..."
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Cited by 5 (3 self)
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Let E be an elliptic curve defined over Q. Let Γ be a free subgroup of rank r of E(Q). For any prime p of good reduction, let Γp be the reduction of Γ modulo p and Ep be the reduction of E modulo p. We prove that if E has CM then for all but o(x / log x) of primes p ≤ x, Γp  ≥ p r r+2 +ɛ(p), where ɛ(p) is any function of p such that ɛ(p) → 0 as p → ∞. This is a consequence of two other results. Denote by Np the cardinality of Ep(Fp), where Fp is a finite field of p elements. Then for any δ> 0, the set of primes p for which Np has a divisor in the range (pδ−ɛ(p), pδ+ɛ(p) ) has density zero. Moreover, the set of primes p for which Γp  < p r r+2 −ɛ(p) has density zero.
Uniform distribution of fractional parts related to pseudoprimes
, 2005
"... We estimate exponential sums with the Fermatlike quotients fg(n) = gn−1 − 1 n and hg(n) = gn−1 − 1 P(n) where g and n are positive integers, n is composite, and P(n) is the largest prime factor of n. Clearly, both fg(n) and hg(n) are integers if n is a Fermat pseudoprime to base g, and if n is a ..."
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Cited by 5 (4 self)
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We estimate exponential sums with the Fermatlike quotients fg(n) = gn−1 − 1 n and hg(n) = gn−1 − 1 P(n) where g and n are positive integers, n is composite, and P(n) is the largest prime factor of n. Clearly, both fg(n) and hg(n) are integers if n is a Fermat pseudoprime to base g, and if n is a Carmichael number this is true for all g coprime to n. Nevertheless, our bounds imply that the fractional parts {fg(n)} and {hg(n)} are uniformly distributed, on average over g for fg(n), and individually for hg(n). We also obtain similar results with the functions ˜ fg(n) = gfg(n) and ˜ hg(n) = ghg(n). AMS Subject Classification: 11L07, 11N37, 11N60 1
An improvement for the large sieve for square moduli
 J. Number Theory
"... Abstract. We establish a result on the large sieve with square moduli. These bounds improve recent results ..."
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Abstract. We establish a result on the large sieve with square moduli. These bounds improve recent results
AN EXPLICIT SIEVE BOUND AND SMALL VALUES OF σ(φ(m))
"... Abstract. We prove an explicit sieve upper bound based on the large sieve of Montgomery and Vaughan [MV], and apply it to show that σ(φ(m)) � m/39.4 for all positive integers m. 1. ..."
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Abstract. We prove an explicit sieve upper bound based on the large sieve of Montgomery and Vaughan [MV], and apply it to show that σ(φ(m)) � m/39.4 for all positive integers m. 1.
LARGE SIEVE INEQUALITIES WITH QUADRATIC AMPLITUDES
, 2005
"... Abstract. In this paper, we develop a large sieve type inequality with quadratic amplitude. We use the double large sieve to establish nontrivial bounds. 1. Introduction and Statements of the Results The large sieve was an idea originated by Yu. V. Linnik [11] in 1941. He also made application to d ..."
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Abstract. In this paper, we develop a large sieve type inequality with quadratic amplitude. We use the double large sieve to establish nontrivial bounds. 1. Introduction and Statements of the Results The large sieve was an idea originated by Yu. V. Linnik [11] in 1941. He also made application to distributions of quadratic nonresidues. Since then, the idea has been refined and perfected by many. We denote ‖x ‖ = min k∈Z x − k  for x ∈ R. A set of real numbers {xk} is said to be δspaced modulo 1 if