## Functiones et Approximatio Commentarii Mathematici

### Some problems of analytic number theory on arithmetic semigroups

#### Abstract

Let $\mathcal{E}$ be a set of primes with density $\tau > 0$ in the set of primes. Write $\mathcal{A}$ for the set of positive integers composed solely of primes from $\mathcal{E}$. We discuss the distribution of the integers from $\mathcal{A}$ in short intervals, and whether for fixed $k \in \mathbb{Z}$ there are solutions to $m+k = p$ with $m \in \mathcal{A}$, where $p$ denotes a prime, or $m+k=n$ where $n$ has a large prime factor ($>n^{\xi}$ for $\xi > \tfrac{1}{2}$

#### Article information

Source
Funct. Approx. Comment. Math. Volume 38, Number 1 (2008), 21-39.

Dates
First available in Project Euclid: 18 December 2008

Permanent link to this document
https://projecteuclid.org/euclid.facm/1229624649

Digital Object Identifier
doi:10.7169/facm/1229624649

Mathematical Reviews number (MathSciNet)
MR2433786

Zentralblatt MATH identifier
1231.11114

#### Citation

Harman, Glyn; Matomäki, Kaisa. Some problems of analytic number theory on arithmetic semigroups. Funct. Approx. Comment. Math. 38 (2008), no. 1, 21--39. doi:10.7169/facm/1229624649. https://projecteuclid.org/euclid.facm/1229624649

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