Functiones et Approximatio Commentarii Mathematici

A note on algebraic integers with prescribed factorization properties in short intervals

Jerzy Kaczorowski

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Abstract

We study the distribution of algebraic integers with prescribed factorization properties in short intervals and prove that for a large class of such numbers from a fixed algebraic number field $K$ with a non-trivial class group, every interval of the form $(x, x+x^{\theta})$ with a fixed $\theta >1/2$ contains absolute value of the norm of such algebraic integer provided $x\geq x_0$. The constant $x_0$ effectively depends on $K$ and $\theta$.

Article information

Source
Funct. Approx. Comment. Math., Volume 40, Number 1 (2009), 151-154.

Dates
First available in Project Euclid: 30 March 2009

Permanent link to this document
https://projecteuclid.org/euclid.facm/1238418805

Digital Object Identifier
doi:10.7169/facm/1238418805

Mathematical Reviews number (MathSciNet)
MR2527636

Zentralblatt MATH identifier
1234.11152

Subjects
Primary: 11R27: Units and factorization
Secondary: 11R42: Zeta functions and $L$-functions of number fields [See also 11M41, 19F27] 11R45: Density theorems 11N25: Distribution of integers with specified multiplicative constraints

Keywords
Factorization in algebraic number fields short intervals unique factorization

Citation

Kaczorowski, Jerzy. A note on algebraic integers with prescribed factorization properties in short intervals. Funct. Approx. Comment. Math. 40 (2009), no. 1, 151--154. doi:10.7169/facm/1238418805. https://projecteuclid.org/euclid.facm/1238418805


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References

  • A. Geroldinger, F. Halter-Koch, Non-Unique Factorizations. Algebraic, Combinatorial and Analytic Theory, Pure and Applied Mathematics (Boca Raton), 278. Chapman & Hall/CRC, Boca Raton, FL, 2006.
  • J. Kaczorowski, Irreducible algebraic integers in short intervals, to appear in Math. Ann.
  • W. Narkiewicz, Elementary and analytic theory of algebraic numbers, Springer-Verlag, Berlin, Heidelberg, 2004.