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The primes contain arbitrarily long arithmetic progressions
 Ann. of Math
"... Abstract. We prove that there are arbitrarily long arithmetic progressions of primes. ..."
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Cited by 150 (26 self)
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Abstract. We prove that there are arbitrarily long arithmetic progressions of 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.
200?), Small gaps between products of two primes
 arXiv.math.NT/0609615. GAPS BETWEEN ALMOST PRIMES 23
"... As an approximation to the twin prime conjecture it was proved in [11] that (1.1) liminf n→∞ pn+1 − pn ..."
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Cited by 10 (3 self)
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As an approximation to the twin prime conjecture it was proved in [11] that (1.1) liminf n→∞ pn+1 − pn
The number of solutions of Φ(x) = m
"... An old conjecture of Sierpiński asserts that for every integer k � 2, there is a number m for which the equation φ(x) = m has exactly k solutions. Here φ is Euler’s totient function. In 1961, Schinzel deduced this conjecture from his Hypothesis H. The purpose of this paper is to present an uncondit ..."
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Cited by 9 (2 self)
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An old conjecture of Sierpiński asserts that for every integer k � 2, there is a number m for which the equation φ(x) = m has exactly k solutions. Here φ is Euler’s totient function. In 1961, Schinzel deduced this conjecture from his Hypothesis H. The purpose of this paper is to present an unconditional proof of Sierpiński’s conjecture. The proof uses many results from sieve theory, in particular the famous theorem of Chen.
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.
Checking the odd Goldbach conjecture up to 10 20
 Math. Comp
, 1998
"... Abstract. Vinogradov’s theorem states that any sufficiently large odd integer is the sum of three prime numbers. This theorem allows us to suppose the conjecture that this is true for all odd integers. In this paper, we describe the implementation of an algorithm which allowed us to check this conje ..."
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Cited by 7 (1 self)
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Abstract. Vinogradov’s theorem states that any sufficiently large odd integer is the sum of three prime numbers. This theorem allows us to suppose the conjecture that this is true for all odd integers. In this paper, we describe the implementation of an algorithm which allowed us to check this conjecture up to 10 20. 1.
On the Size of the First Factor of the Class Number of a Cyclotomic Field
, 1990
"... We show that Kummer's conjectured asymptotic estimate for the size of the first factor of the class number of a cyclotomic field is untrue under the assumption of two wellknown and widely believed conjectures of analytic number theory. 1. Introduction In 1850 Kummer [13] published a review of the ..."
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Cited by 6 (2 self)
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We show that Kummer's conjectured asymptotic estimate for the size of the first factor of the class number of a cyclotomic field is untrue under the assumption of two wellknown and widely believed conjectures of analytic number theory. 1. Introduction In 1850 Kummer [13] published a review of the main results that he and others had discovered about cyclotomic fields. In this elegant report he claimed that he had found an explicit "law for the asymptotic growth" of h 1 (p), the socalled first factor of the class number of the cyclotomic field, and would provide a proof elsewhere. This proof never appeared and we believe that Kummer's claim is incorrect. More precisely, let p denote any odd prime, let h(p) be the class number of the cyclotomic field Q(i p ) (where i p is a primitive pth root of unity) and h 2 (p) be the class number of the real subfield Q(i p +i \Gamma1 p ). Kummer proved that the ratio h 1 (p) = h(p)=h 2 (p) is an integer which he called the first factor of the ...
Obstructions to uniformity, and arithmetic patterns in the primes, preprint
"... Abstract. In this expository article, we describe the recent approach, motivated by ergodic theory, towards detecting arithmetic patterns in the primes, and in particular establishing in [26] that the primes contain arbitrarily long arithmetic progressions. One of the driving philosophies is to iden ..."
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Cited by 5 (3 self)
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Abstract. In this expository article, we describe the recent approach, motivated by ergodic theory, towards detecting arithmetic patterns in the primes, and in particular establishing in [26] that the primes contain arbitrarily long arithmetic progressions. One of the driving philosophies is to identify precisely what the obstructions could be that prevent the primes (or any other set) from behaving “randomly”, and then either show that the obstructions do not actually occur, or else convert the obstructions into usable structural information on the primes. 1.
Average Order in Cyclic Groups
"... For each natural number n we determine the average order #(n) of the elements in a cyclic group of order n. We show that a large fraction of the contribution to #(n) comes from the #(n) primitive elements of order n. It is therefore of interest to study also the function #(n) = #(n)/#(n). We determi ..."
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Cited by 3 (1 self)
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For each natural number n we determine the average order #(n) of the elements in a cyclic group of order n. We show that a large fraction of the contribution to #(n) comes from the #(n) primitive elements of order n. It is therefore of interest to study also the function #(n) = #(n)/#(n). We determine the mean behavior of #, #, 1/#, and also consider these functions in the multiplicative groups of finite fields.
TERNARY GOLDBACH PROBLEM FOR THE SUBSETS OF PRIMES WITH POSITIVE RELATIVE DENSITIES
, 2007
"... Abstract. Let P denote the set of all primes. Suppose that P1, P2, P3 are three subsets of P with d P (P1) + d P (P2) + d P (P3)> 2, where d P (Pi) is the lower density of Pi relative to P. We prove that for sufficiently large odd integer n, there exist pi ∈ Pi such that n = p1 + p2 + p3. 1. ..."
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Cited by 3 (3 self)
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Abstract. Let P denote the set of all primes. Suppose that P1, P2, P3 are three subsets of P with d P (P1) + d P (P2) + d P (P3)> 2, where d P (Pi) is the lower density of Pi relative to P. We prove that for sufficiently large odd integer n, there exist pi ∈ Pi such that n = p1 + p2 + p3. 1.