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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.
Restriction theory of Selberg’s sieve, with applications, to appear, Journal de Theorie de Nombres de Bordeaux
"... Abstract. The Selberg sieve provides majorants for certain arithmetic sequences, such as the primes and the twin primes. We prove an L 2 –L p restriction theorem for majorants of this type. An immediate application is to the estimation of exponential sums over prime ktuples. Let a1,..., ak and b1,. ..."
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Cited by 14 (7 self)
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Abstract. The Selberg sieve provides majorants for certain arithmetic sequences, such as the primes and the twin primes. We prove an L 2 –L p restriction theorem for majorants of this type. An immediate application is to the estimation of exponential sums over prime ktuples. Let a1,..., ak and b1,...,bk be positive integers. Write h(θ): = ∑ n∈X e(nθ), where X is the set of all n � N such that the numbers a1n + b1,..., akn + bk are all prime. We obtain upper bounds for ‖h ‖ L p (T), p> 2, which are (conditionally on the prime tuple conjecture) of the correct order of magnitude. As a second application we deduce from Chen’s theorem, Roth’s theorem, and a transference principle that there are infinitely many arithmetic progressions p1 < p2 < p3 of primes, such that pi + 2 is either a prime or a product of two primes for each i = 1, 2, 3.
On the second moment for primes in an arithmetic progression, Acta Arithmetica C.1
, 2001
"... Abstract. Assuming the Generalized Riemann Hypothesis, we obtain a lower bound within a constant factor of the conjectured asymptotic result for the second moment for primes in an individual arithmetic progression in short intervals. Previous results were averaged over all progression of a given mod ..."
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Cited by 3 (2 self)
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Abstract. Assuming the Generalized Riemann Hypothesis, we obtain a lower bound within a constant factor of the conjectured asymptotic result for the second moment for primes in an individual arithmetic progression in short intervals. Previous results were averaged over all progression of a given modulus. The method uses a short divisor sum approximation for the von Mangoldt function, together with some new results for binary correlations of this divisor sum approximation in arithmetic progressions. 1. Introduction and Statement of
Journal de Théorie des Nombres
, 2005
"... Restriction theory of the Selberg sieve, with applications ..."
A Multiple Sum Involving the Möbius Function
, 2003
"... 1. Introduction. The aim of the present article is to discuss the asymptotics of the quantity Mk(z) = ∑ µ(d1) · · · µ(d2k), k ≥ 1, (1.1) ..."
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1. Introduction. The aim of the present article is to discuss the asymptotics of the quantity Mk(z) = ∑ µ(d1) · · · µ(d2k), k ≥ 1, (1.1)
A NOTE ON TWIN PRIMES
, 2005
"... ABSTRACT. We relate the twin prime conjecture to corresponding conjectures for a short divisor sum which approximates primes. The twin prime conjecture states that there are infinitely many pairs of primes differing by two. More generally we expect there will be infinitely many pairs of primes with ..."
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ABSTRACT. We relate the twin prime conjecture to corresponding conjectures for a short divisor sum which approximates primes. The twin prime conjecture states that there are infinitely many pairs of primes differing by two. More generally we expect there will be infinitely many pairs of primes with difference k, for any fixed even integer k. Let ƒ.n / be the von Mangoldt function defined to be log p if n D pm, m 1, and zero otherwise. Then a quantitative version of the general twin prime conjecture is that, as N!1, NX (1) ƒ.n/ƒ.n C k / D.S.k / C o.1//N; nD1 where S.k / is the singular series given by 8