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31
The Spectrum of Multiplicative Functions
"... this paper is to understand the spectrum. Although we can determine the spectrum explicitly only in one interesting case (where S = [\Gamma1; 1]), we are able, in general, to qualitatively describe it and obtain some of its geometric structure. For example, qualitatively, the spectrum may be describ ..."
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Cited by 20 (12 self)
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this paper is to understand the spectrum. Although we can determine the spectrum explicitly only in one interesting case (where S = [\Gamma1; 1]), we are able, in general, to qualitatively describe it and obtain some of its geometric structure. For example, qualitatively, the spectrum may be described in terms of Euler products and solutions to certain integral equations. Geometrically, we can always determine the boundary points of the spectrum (that is, the elements of \Gamma(S) " T) and show that the spectrum is connected. Moreover we can bound the spectrum, and make conjectures about some of its properties, though we have no precise idea of what it usually looks like. We begin with a few immediate consequences of our definition:
On quantum ergodicity for linear maps of the torus
 COMM. MATH. PHYS
, 1999
"... We prove a strong version of quantum ergodicity for linear hyperbolic maps of the torus (“cat maps”). We show that there is a density one sequence of integers so that as N tends to infinity along this sequence, all eigenfunctions of the quantum propagator at inverse Planck constant N are uniformly ..."
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Cited by 15 (3 self)
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We prove a strong version of quantum ergodicity for linear hyperbolic maps of the torus (“cat maps”). We show that there is a density one sequence of integers so that as N tends to infinity along this sequence, all eigenfunctions of the quantum propagator at inverse Planck constant N are uniformly distributed. A key step in the argument is to show that for a hyperbolic matrix in the modular group, there is a density one sequence of integers N for which its order (or period) modulo N is somewhat larger than √ N.
An asymptotic formula for the number of smooth values of a polynomial
 J. Number Theory
, 1999
"... Integers without large prime factors, dubbed smooth numbers, are by now firmly established as a useful and versatile tool in number theory. More than being simply a property of numbers that is conceptually dual to primality, smoothness has played a major role in the proofs of many results, from mult ..."
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Cited by 13 (1 self)
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Integers without large prime factors, dubbed smooth numbers, are by now firmly established as a useful and versatile tool in number theory. More than being simply a property of numbers that is conceptually dual to primality, smoothness has played a major role in the proofs of many results, from multiplicative questions to Waring’s problem to complexity
Chebyshev’s bias for composite numbers with restricted prime divisors
 Math. Comp
, 2005
"... Abstract. Let π(x; d, a) denote the number of primes p ≤ x with p ≡ a(mod d). Chebyshev’s bias is the phenomenon for which “more often” π(x; d, n)>π(x; d, r), than the other way around, where n is a quadratic nonresidue mod d and r is a quadratic residue mod d. Ifπ(x; d, n) ≥ π(x; d, r) for ever ..."
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Cited by 12 (6 self)
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Abstract. Let π(x; d, a) denote the number of primes p ≤ x with p ≡ a(mod d). Chebyshev’s bias is the phenomenon for which “more often” π(x; d, n)>π(x; d, r), than the other way around, where n is a quadratic nonresidue mod d and r is a quadratic residue mod d. Ifπ(x; d, n) ≥ π(x; d, r) for every x up to some large number, then one expects that N(x; d, n) ≥ N(x; d, r) for every x. Here N(x; d, a) denotes the number of integers n ≤ x such that every prime divisor p of n satisfies p ≡ a(mod d). In this paper we develop some tools to deal with this type of problem and apply them to show that, for example, N(x;4, 3) ≥ N(x;4, 1) for every x. In the process we express the socalled second order LandauRamanujan constant as an infinite series and show that the same type of formula holds for a much larger class of constants. 1.
Decay of meanvalues of multiplicative functions
 Canad. J. Math
"... Given a multiplicative function f with f(n)  ≤ 1 for all n, we are concerned with obtaining explicit upper bounds on the meanvalue 1 ..."
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Cited by 7 (4 self)
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Given a multiplicative function f with f(n)  ≤ 1 for all n, we are concerned with obtaining explicit upper bounds on the meanvalue 1
An uncertainty principle for arithmetic sequences
, 2004
"... Analytic number theorists usually seek to show that sequences which appear naturally in arithmetic are “welldistributed ” in some appropriate sense. In various discrepancy problems, combinatorics researchers have analyzed limitations to equidistribution, as have Fourier analysts when working with ..."
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Cited by 6 (3 self)
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Analytic number theorists usually seek to show that sequences which appear naturally in arithmetic are “welldistributed ” in some appropriate sense. In various discrepancy problems, combinatorics researchers have analyzed limitations to equidistribution, as have Fourier analysts when working with the “uncertainty principle”. In this article we find that these ideas have a natural setting in the analysis of distributions of sequences in analytic number theory, formulating a general principle, and giving several examples.
Squarefree Values of the Carmichael Function
 J. NUM. THEORY
, 2003
"... We obtain an asymptotic formula for the number of squarefree values among p 1; for primes ppx; and we apply it to derive the following asymptotic formula for LðxÞ; the number of squarefree values of the Carmichael function lðnÞ for 1pnpx; LðxÞ ðk þ oð1ÞÞ x ln 1 a x; where a 0:37395y is the Artin ..."
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Cited by 5 (3 self)
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We obtain an asymptotic formula for the number of squarefree values among p 1; for primes ppx; and we apply it to derive the following asymptotic formula for LðxÞ; the number of squarefree values of the Carmichael function lðnÞ for 1pnpx; LðxÞ ðk þ oð1ÞÞ x ln 1 a x; where a 0:37395y is the Artin constant, and k 0:80328y is another absolute constant.