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Prime numbers of the form p = m²+n²+1 in short intervals

100%

Matomäki K.

Acta Arithmetica

2007

tom 128

nr 2

193-200

A note on primes of the form p=aq² + 1

100%

Matomäki K.

Acta Arithmetica

2009

tom 137

nr 2

133-137

Sums of positive density subsets of the primes

100%

Matomäki K.

Acta Arithmetica

2013

tom 159

nr 3

201-225

We show that if A and B are subsets of the primes with positive relative lower densities α and β, then the lower density of A+B in the natural numbers is at least $(1-o(1))α/(e^{γ} log log (1/β))$, which is asymptotically best possible. This improves results of Ramaré and Ruzsa and of Chipeniuk and Hamel. As in the latter work, the problem is reduced to a similar problem for subsets of $ℤ*_m$ using techniques of Green and Green-Tao. Concerning this new problem we show that, for any square-free m and any $A, B ⊆ ℤ*_m$ of densities α and β, the density of A+B in $ℤ_m$ is at least $(1-o(1))α/(e^{γ} log log (1/β))$, which is asymptotically best possible when m is a product of small primes. We also discuss an inverse question.

Ograniczanie wyników

3 Acta Arithmetica

3 Matomäki K.

1 2013

1 2009

1 2007

Wyniki wyszukiwania

Prime numbers of the form p = m²+n²+1 in short intervals

A note on primes of the form p=aq² + 1

Sums of positive density subsets of the primes