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Higher correlations of divisor sums related to primes, II: Variations of . . .
, 2007
"... We calculate the triple correlations for the truncated divisor sum λR(n). The λR(n) behave over certain averages just as the prime counting von Mangoldt function Λ(n) does or is conjectured to do. We also calculate the mixed (with a factor of Λ(n)) correlations. The results for the moments up to the ..."
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Cited by 28 (6 self)
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We calculate the triple correlations for the truncated divisor sum λR(n). The λR(n) behave over certain averages just as the prime counting von Mangoldt function Λ(n) does or is conjectured to do. We also calculate the mixed (with a factor of Λ(n)) correlations. The results for the moments up to the third degree, and therefore the implications for the distribution of primes in short intervals, are the same as those we obtained (in the first paper with this title) by using the simpler approximation ΛR(n). However, when λR(n) is used, the error in the singular series approximation is often much smaller than what ΛR(n) allows. Assuming the Generalized Riemann Hypothesis (GRH) for Dirichlet Lfunctions, we obtain an Ω±result for the variation of the error term in the prime number theorem. Formerly, our knowledge under GRH was restricted to Ωresults for the absolute value of this variation. An important ingredient in the last part of this work is a recent result due to Montgomery and Soundararajan which makes it possible for us to dispense with a large error term in the evaluation of a certain singular series average. We believe that our results on the sums λR(n) and ΛR(n) can be employed in diverse problems concerning primes.
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 19 (11 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:
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
Lattice Points on Circles and the Discrete Velocity Model for the Boltzmann Equation
, 2004
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On the Supremum of Random Dirichlet Polynomials
, 2007
"... Abstract: We study the supremum of some random Dirichlet polynomials DN(t) = ∑N n=2 εndnn−σ−it, where (εn) is a sequence of independent Rademacher random variables, the weights (dn) are multiplicative and 0 ≤ σ < 1/2. The particular attention is given to the polynomials n∈Eτ εnn−σ−it, Eτ = {2 ≤ n ≤ ..."
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Cited by 3 (3 self)
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Abstract: We study the supremum of some random Dirichlet polynomials DN(t) = ∑N n=2 εndnn−σ−it, where (εn) is a sequence of independent Rademacher random variables, the weights (dn) are multiplicative and 0 ≤ σ < 1/2. The particular attention is given to the polynomials n∈Eτ εnn−σ−it, Eτ = {2 ≤ n ≤ N: P +(n) ≤ pτ}, P +(n) being the largest prime divisor of n. We obtain sharp upper and lower bounds for supremum expectation that extend the optimal estimate of HalászQueffélec
Counting the number of solutions to the ErdősStraus equation on unit fractions
"... Abstract. For any positive integer n, let f(n) denote the number of solutions to the Diophantine equation 4 1 1 1 = + + with x, y, z positive integers. The ErdősStraus conjecture asserts that n x y z f(n)> 0 for every n � 2. To solve this conjecture, it suffices without loss of generality to consid ..."
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Cited by 2 (0 self)
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Abstract. For any positive integer n, let f(n) denote the number of solutions to the Diophantine equation 4 1 1 1 = + + with x, y, z positive integers. The ErdősStraus conjecture asserts that n x y z f(n)> 0 for every n � 2. To solve this conjecture, it suffices without loss of generality to consider the case when n is a prime p. In this paper we consider the question of bounding the sum ∑ p<N f(p) asymptotically as N → ∞, where p ranges over primes. Our main result establishes the asymptotic upper and lower bounds N log 2 N ≪ ∑ f(p) ≪ N log 2 N log log N. p�N In particular, f(p) = Oδ(log3 p log log p) for a subset of primes of density δ arbitrarily close to 1. Also, for a subset of the primes with density 1 the following lower bound holds: f(p) ≫ (log p) 0.549. These upper and lower bounds show that a typical prime has a small number of solutions to the ErdősStraus Diophantine equation; small, when compared with other additive problems, like Waring’s problem. We establish several more results on f and related quantities, for instance the bound f(p) ≪ p 3 5 +O ( 1 log log p) for all primes p. Eventually we prove lower bounds for the number fm,k(n) of solutions of m n
QUANTUM UNIQUE ERGODICITY AND NUMBER THEORY
"... In this course I will describe recent progress on the “Quantum Unique Ergodicity ” conjecture of Rudnick and Sarnak in a special arithmetic situation. To explain what this conjecture is about, let H denote the upper half plane {x + iy: y> 0}. The group SL2(R) acts on H by ..."
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In this course I will describe recent progress on the “Quantum Unique Ergodicity ” conjecture of Rudnick and Sarnak in a special arithmetic situation. To explain what this conjecture is about, let H denote the upper half plane {x + iy: y> 0}. The group SL2(R) acts on H by
Decay Of MeanValues Of Multiplicative Functions
"... Introduction Given a multiplicative function f with jf(n)j 1 for all n, we are concerned with obtaining explicit upper bounds on the meanvalue 1 x j P nx f(n)j. Ideally, one would like to give a bound for this meanvalue which depends only on a knowledge of f(p) for primes p. To illustrate ..."
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Introduction Given a multiplicative function f with jf(n)j 1 for all n, we are concerned with obtaining explicit upper bounds on the meanvalue 1 x j P nx f(n)j. Ideally, one would like to give a bound for this meanvalue which depends only on a knowledge of f(p) for primes p. To illustrate what we mean, we recall a pioneering result of E. Wirsing [16]. Throughout, we put \Theta(f; x) := Y px i 1 + f(p) p + f(p 2 ) p 2 + : : : ji 1 \Gamma 1 p j : A. Wintner [15]