Scharfe untere Abschätzung für die Anzahlfunktion der B-Zwillinge

Indlekofer, Karl-Heinz

doi:10.4064/aa-26-2-207-212

Cited by 15 publications

(6 citation statements)

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“…As examples of sufficiently sifted sequences, let us mention the sequence of integers (~ = 0), the sequence of primes (rc = 1) , the sequence of integers that are sums of two squares (x = ~), the sequence of integers n that are sums of two coprime squares and such that n+1 also has this property (x = 1, cf. [10]) or the sequence of those prime numbers p such that p can be written as p = + n2 with (m, n) = 1 (x = ~, cf. [11] and also [7] for related sequences).…”

Section: Introductionmentioning

confidence: 99%

Additive properties of dense subsets of sifted sequences

Ramaré¹,

Ruzsa²

2001

Journal De Théorie Des Nombres De Bordeaux

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Section: Introductionmentioning

confidence: 99%

Additive properties of dense subsets of sifted sequences

Ramaré¹,

Ruzsa²

2001

Journal De Théorie Des Nombres De Bordeaux

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“…The sequences C and D above are not known to have an asymptotic density, though they are believed to possess one. We know however that (see [13] for the lower bound) c ≤ X, c ∈ C X/ log X and that (see [14] for the lower bound)…”

Section: Introduction and Some Resultsmentioning

confidence: 99%

A note on additive properties of dense subsets of sifted sequences

Ramaré¹

2013

Bulletin of the London Mathematical Society

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In this paper we show that if A is a subset of the primes with positive lower relative density δ then A + A must have positive lower density at least C1δ/ log log(1/δ) in the natural numbers. Our argument uses the techniques developed by the author and I.Ruzsa in their work on additive properties of dense subsequences of sufficiently sifted sequences. The result is optimal and improves on recent work of K.Chipeniuk & M.Hamel. We continue by proving several similar results, by successively replacing the sequence of primes by the sequence of sums of two squares, by the sequence of those integers n that are such that n and n + 1 are both a sum of two squares and finally by the sequence of primes p that are such that p + 1 is a sum of two squares. The second part of this paper contains a heuristical argument that leads to several conjectures concerning the existence of k-term arithmetic progressions within these sequences. We conclude with some conjectures belonging to the Ramsey part of additive number theory.

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“…The number of such « < x is :» x/(logx)5/8 (loglogx)5, and the number of such « < x for which F(n -4) = F(« -2) = F(n) = F(n + 2) = F(n + 4) = 1 fails is 0( x/log x). This method runs into difficulties when six or more consecutive even numbers are considered, but perhaps the methods of C. Hooley [13] and K.-H. Indlekofer [14] would be of use. Thanks are due to Helmut Maier for the suggestion.…”

Section: Corollarymentioning

confidence: 99%