Functiones et Approximatio Commentarii Mathematici

Some problems of analytic number theory on arithmetic semigroups

Glyn Harman and Kaisa Matomäki

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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

Subjects
Primary: 11N25: Distribution of integers with specified multiplicative constraints
Secondary: 11N36: Applications of sieve methods

Keywords
greatest prime factors distribution in short intervals

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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References

  • R.C. Baker and G. Harman, The Brun-Titchmarsh Theorem on Average, Progress in Mathematics 138, Birkhäuser, Boston, 1996.
  • R.C. Baker and G. Harman, Shifted primes without large prime factors, Acta Arith. 83 (1998), 331--361.
  • R.C. Baker, G. Harman, and J. Pintz, The difference between consecutive primes II, Proc. London Math. Soc. (3) 83 (2001), 532--562.
  • A. Beurling, Analyse de la loi asymptotique de la distribution des nombres premiers géneralisés, Acta Math. 68 (1937) 255-291.
  • E. Bombieri, J.B. Friedlander, and H. Iwaniec, Primes in Arithmetic Progressions to Large Moduli. II, Math. Ann. 227 (1987), 361-393.
  • H. Davenport. Multiplicative Number Theory, 2nd ed. (revised by H. L. Montgomery), Springer, New York, 1980.
  • H. Delange, Sur la distribution des entiers ayant certaines propriétés, Ann. Sci. Ecole Norm. Sup. (3) 73 (1956), 15--74.
  • C. Elsholtz, Additive decomposability of multiplicatively defined sets, Funct. Approx. 35 (2006), 61--77.
  • C. Elsholtz, Multiplicative decomposability of additively shifted primes, to appear.
  • G. Greaves, Sieves in Number Theory, Springer-Verlag, Berlin 2001.
  • H. Halberstam and H.-E. Richert, Sieve Methods, Academic Press, London, 1974.
  • G. Harman, On the greatest prime factor of $p-1$ with effective constants, Math. Comp. 74 (2005), 2035--2041.
  • G. Harman, Prime-Detecting Sieves, LMS Monographs (New Series) 33, Princeton University Press, 2007.
  • A. Hildebrand and G. Tenenbaum, Integers without large prime factors, Journal de Théorie des Nombres de Bordeaux, 5 (1993), 411--484.
  • M.N. Huxley, On the difference between consecutive primes, Invent. Math. 15 (1972), 164--170.
  • H. Iwaniec, Primes of the type $\phi(x,\,y)+A$ where $\phi$ is a quadratic form, Acta Arithmetica 21 (1972), 203--234.
  • H. Iwaniec, Rosser's Sieve, Acta Arithmetica 36 (1980), 171--202.
  • K. Matomäki, Prime numbers of the form $m^2 + n^2 + 1$ in short intervals, Acta Arithmetica 128 (2007), 193--200.
  • H.L. Montgomery and R.C. Vaughan, The large sieve, Mathematika 20 (1973), 119--134.
  • H.L. Mongomery and R.C. Vaughan, Hilbert's Inequality, J. London Math. Soc. (2), 8 (1974), 73 -- 82.
  • A.G. Postnikov, Introduction to Analytic Number Theory, Translations of AMS Monographs 68, American Mathematical Society, Providence, Rhode Island, 1988.
  • C.L. Stewart, On the greatest prime factors of integers of the form $ab+1$, Period. Math. Hungar. 43 (2001), 81--91.
  • G. Tenenbaum, Cribler les entiers sans grand facteur premier, Philos. Trans. Roy. Soc. London Ser. A 345 (1993), 377--384.
  • E.C. Titchmarsh, The Theory of the Riemann Zeta-function, (revised D.R. Heath-Brown), Oxford 1986.
  • R.C. Vaughan, An elementary method in prime number theory, Acta Arith. 37 (1980), 111--115.
  • E. Wirsing, Über die Zahlen, deren Primteiler einer gegebenen Menge angehören, Arch. Math. 7 (1956), 263--272.