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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.
On The Lattice Point Problem For Ellipsoids
, 1994
"... For kdimensional ellipsoids E with k 9 we show that the number of lattice points in rE is approximated by the volume of rE up to an error of order O(r k\Gamma2 ), as r tends to infinity. The estimate is best possible and uniform in the class of ellipsoids with bounded ratios of lengths of axes ..."
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Cited by 15 (6 self)
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For kdimensional ellipsoids E with k 9 we show that the number of lattice points in rE is approximated by the volume of rE up to an error of order O(r k\Gamma2 ), as r tends to infinity. The estimate is best possible and uniform in the class of ellipsoids with bounded ratios of lengths of axes. It improves the classical estimate O(r k\Gamma2k=(k+1) ) due to Landau (1915).
Yıldırım, Small gaps between primes or almost primes
"... Abstract. Let pn denote the nth prime. Goldston, Pintz, and Yıldırım recently proved that (pn+1 − pn) lim inf =0. n→ ∞ log pn We give an alternative proof of this result. We also prove some corresponding results for numbers with two prime factors. Let qn denote the nth number that is a product of ex ..."
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Cited by 8 (2 self)
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Abstract. Let pn denote the nth prime. Goldston, Pintz, and Yıldırım recently proved that (pn+1 − pn) lim inf =0. n→ ∞ log pn We give an alternative proof of this result. We also prove some corresponding results for numbers with two prime factors. Let qn denote the nth number that is a product of exactly two distinct primes. We prove that lim inf n→ ∞ (qn+1 − qn) ≤ 26. If an appropriate generalization of the ElliottHalberstam Conjecture is true, then the above bound can be improved to 6. 1.
An invitation to additive prime number theory
, 2004
"... The main purpose of this survey is to introduce the inexperienced reader to additive prime number theory and some related branches of analytic number theory. We state the main problems in the field, sketch their history and the basic machinery used to study them, and try to give a representative sam ..."
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Cited by 4 (0 self)
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The main purpose of this survey is to introduce the inexperienced reader to additive prime number theory and some related branches of analytic number theory. We state the main problems in the field, sketch their history and the basic machinery used to study them, and try to give a representative sample of the directions of current research.
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
Sieve Methods
"... Sieve methods have had a long and fruitful history. The sieve of Eratosthenes (around 3rd century B.C.) was a device to generate prime numbers. Later Legendre used it in his studies of the prime number counting function π(x). Sieve methods bloomed and became a topic of intense investigation after th ..."
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Sieve methods have had a long and fruitful history. The sieve of Eratosthenes (around 3rd century B.C.) was a device to generate prime numbers. Later Legendre used it in his studies of the prime number counting function π(x). Sieve methods bloomed and became a topic of intense investigation after the pioneering work of Viggo Brun (see [Bru16],[Bru19], [Bru22]). Using his formulation of the sieve Brun proved, that the sum
unknown title
"... On the greatest prime factor of integers of the form ab + 1 by C.L. Stewart ∗ to Professor András Sárközy on the occasion of his sixtieth birthday Abstract: Let N be a positive integer and let A and B be dense subsets of {1,..., N}. The purpose of this paper is to establish a good lower bound for th ..."
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On the greatest prime factor of integers of the form ab + 1 by C.L. Stewart ∗ to Professor András Sárközy on the occasion of his sixtieth birthday Abstract: Let N be a positive integer and let A and B be dense subsets of {1,..., N}. The purpose of this paper is to establish a good lower bound for the greatest prime factor of ab + 1 as a and b run over the elements of A and B respectively.
A weighted Turán sieve method
, 2006
"... We develop a weighted Turán sieve method and applied it to study the number of distinct prime divisors of f(p) where p is a prime and f(x) a polynomial with integer coefficients. ..."
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We develop a weighted Turán sieve method and applied it to study the number of distinct prime divisors of f(p) where p is a prime and f(x) a polynomial with integer coefficients.