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14
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 pair correlation of the zeros of the Riemann zetafunction
 Proc. London Math. Soc
"... In 1972 Montgomery [20, 21] introduced a new method for studying the zeros of the Riemann zetafunction. One of his main accomplishments was to determine partially the pair correlation of zeros, and to apply his results to obtain new information on multiplicity of zeros and gaps between zeros. Perha ..."
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Cited by 8 (3 self)
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In 1972 Montgomery [20, 21] introduced a new method for studying the zeros of the Riemann zetafunction. One of his main accomplishments was to determine partially the pair correlation of zeros, and to apply his results to obtain new information on multiplicity of zeros and gaps between zeros. Perhaps more
Variance of distribution of primes in residue classes
 Quart. J. Math. Oxford Ser
, 1996
"... THE prime number theorem for arithmetic progressions tells us that, for integers a, q s = 1 with (a, q) = 1 we have ..."
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Cited by 6 (3 self)
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THE prime number theorem for arithmetic progressions tells us that, for integers a, q s = 1 with (a, q) = 1 we have
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
Yıldırım, Primes in short segments of arithmetic progressions
 Canad. J. Math
, 1998
"... ABSTRACT. Consider the variance for the number of primes that are both in the interval [y, y + h] for y 2 [x,2x] and in an arithmetic progression of modulus q. We study the total variance obtained by adding these variances over all the reduced residue classes modulo q. Assuming a strong form of the ..."
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Cited by 2 (2 self)
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ABSTRACT. Consider the variance for the number of primes that are both in the interval [y, y + h] for y 2 [x,2x] and in an arithmetic progression of modulus q. We study the total variance obtained by adding these variances over all the reduced residue classes modulo q. Assuming a strong form of the twin prime conjecture and the Riemann Hypothesis one can obtain an asymptotic formula for the total variance in the range when 1 hÛq x 1Û2 è, for anyèÙ0. We show that one can still obtain some weaker asymptotic results assuming the Generalized Riemann Hypothesis (GRH) in place of the twin prime conjecture. In their simplest form, our results are that on GRH the same asymptotic formula obtained with the twin prime conjecture is true for “almost all ” q in the range 1 hÛq h 1Û4 è, that on averaging over q one obtains an asymptotic formula in the extended range 1 hÛq h 1Û2 è, and that there are lower bounds with the correct order of magnitude for all q in the range 1 hÛq x 1Û3 è.
Chebyshev's Bias
, 1994
"... this paper we take a somewhat different point of view in our attempt to analyze Chebyshev's phenomenon and its generalizations, which we call "Chebyshev's bias". Our purpose has been to examine these issues both theoretically and numerically and, in particular, to give numerical values to these bias ..."
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Cited by 1 (0 self)
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this paper we take a somewhat different point of view in our attempt to analyze Chebyshev's phenomenon and its generalizations, which we call "Chebyshev's bias". Our purpose has been to examine these issues both theoretically and numerically and, in particular, to give numerical values to these biases. Let a 1 ; a 2 ; : : : ; a r 2 A q be distinct, and define
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
Mathematisches Forschungsinstitut Oberwolfach Report No. 46/2004 Theory of the Riemann Zeta and Allied Functions
, 2004
"... Introduction by the Organisers This meeting, the second Oberwolfach workshop devoted to zeta functions, was attended by 42 participants representing 16 countries. The scientific program consisted of 32 talks of various lengths and a problem session. In addition, social activities were organised: a h ..."
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Introduction by the Organisers This meeting, the second Oberwolfach workshop devoted to zeta functions, was attended by 42 participants representing 16 countries. The scientific program consisted of 32 talks of various lengths and a problem session. In addition, social activities were organised: a hike in the mountains and piano recitals by Peter