The complex sum of digits function and primes

Jörg M. Thuswaldner

Journal de théorie des nombres de Bordeaux (2000)

  • Volume: 12, Issue: 1, page 133-146
  • ISSN: 1246-7405

Abstract

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Canonical number systems in the ring of gaussian integers ℤ [ i ] are the natural generalization of ordinary q -adic number systems to ℤ [ i ] . It turns out, that each gaussian integer has a unique representation with respect to the powers of a certain base number b . In this paper we investigate the sum of digits function ν b of such number systems. First we prove a theorem on the sum of digits of numbers, that are not divisible by the f -th power of a prime. Furthermore, we establish an Erdös-Kac type theorem for ν b . In all proofs the equidistribution of ν b in residue classes plays a crucial rôle. Starting from this fact we use sieve methods and a version of the model of Kubilius to prove our results.

How to cite

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Thuswaldner, Jörg M.. "The complex sum of digits function and primes." Journal de théorie des nombres de Bordeaux 12.1 (2000): 133-146. <http://eudml.org/doc/248481>.

@article{Thuswaldner2000,
abstract = {Canonical number systems in the ring of gaussian integers $\mathbb \{Z\}[i]$ are the natural generalization of ordinary $q$-adic number systems to $\mathbb \{Z\}[i]$. It turns out, that each gaussian integer has a unique representation with respect to the powers of a certain base number $b$. In this paper we investigate the sum of digits function $\nu _b$ of such number systems. First we prove a theorem on the sum of digits of numbers, that are not divisible by the $f$-th power of a prime. Furthermore, we establish an Erdös-Kac type theorem for $\nu _b$. In all proofs the equidistribution of $\nu _b$ in residue classes plays a crucial rôle. Starting from this fact we use sieve methods and a version of the model of Kubilius to prove our results.},
author = {Thuswaldner, Jörg M.},
journal = {Journal de théorie des nombres de Bordeaux},
keywords = {radix representation; Gaussian integers; sum of digits function; Erdős-Kac Theorem},
language = {eng},
number = {1},
pages = {133-146},
publisher = {Université Bordeaux I},
title = {The complex sum of digits function and primes},
url = {http://eudml.org/doc/248481},
volume = {12},
year = {2000},
}

TY - JOUR
AU - Thuswaldner, Jörg M.
TI - The complex sum of digits function and primes
JO - Journal de théorie des nombres de Bordeaux
PY - 2000
PB - Université Bordeaux I
VL - 12
IS - 1
SP - 133
EP - 146
AB - Canonical number systems in the ring of gaussian integers $\mathbb {Z}[i]$ are the natural generalization of ordinary $q$-adic number systems to $\mathbb {Z}[i]$. It turns out, that each gaussian integer has a unique representation with respect to the powers of a certain base number $b$. In this paper we investigate the sum of digits function $\nu _b$ of such number systems. First we prove a theorem on the sum of digits of numbers, that are not divisible by the $f$-th power of a prime. Furthermore, we establish an Erdös-Kac type theorem for $\nu _b$. In all proofs the equidistribution of $\nu _b$ in residue classes plays a crucial rôle. Starting from this fact we use sieve methods and a version of the model of Kubilius to prove our results.
LA - eng
KW - radix representation; Gaussian integers; sum of digits function; Erdős-Kac Theorem
UR - http://eudml.org/doc/248481
ER -

References

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  1. [1] A.C. Berry, The accuracy of the Gaussian approximation to the sum of independent variates. Trans. Amer. Math. Soc49 (1941), 121-136. Zbl0025.34603MR3498JFM67.0461.01
  2. [2] A.N. Danilov, On sequences of values of additive arithmetical functions defined on the set of ideals of a field K of degree n over the field of rational numbers (russian). Leningrad Gos. Ped. Inst. Ucen. Zap274 (1965), 59-70. Zbl0161.04602MR197430
  3. [3] P.D.T.A. Elliott, Probabilistic number theory i. central limit theorems. Springer, New-York, 1979. Zbl0431.10029
  4. [4] C.G. Esseen, Fourier analysis of distribution functions. A mathematical study of the Laplace-Gaussian law. Acta Math.77 (1945), 1-125. Zbl0060.28705MR14626
  5. [5] A.O. Gelfond, Sur les nombres qui ont des propriétés additives et multiplicatives données. Acta Arithmetica13 (1968), 259-265. Zbl0155.09003MR220693
  6. [6] B. Gittenberger and J.M. Thuswaldner, The moments of the sum of digits function in number fields. Canadian Math. Bull.42 (1999), 68-77. Zbl1011.11009MR1695870
  7. [7] P.J. Grabner, P. Kirschenhofer, and H. Prodinger, The sum- of- digits-function for complex bases. J. London Math. Soc., to appear. Zbl0959.11045MR1624777
  8. [8] G.H. Hardy and E.M. Wright, An introduction to the theory of numbers. 2nd ed., Clarendon Press, Oxford, 1960. Zbl0086.25803MR568909
  9. [9] L.K. Hua, On exponential sums over an algebraic number field. Can. J. Math.3 (1951), no. 1, 44-51. Zbl0042.04301MR47697
  10. [10] Z. Juskys, Limit theorems for additive functions defined on ordered semigroups with a regular norm (russian). Lietuvos Matematikos Rinkinys4 (1964), 565-603. Zbl0151.03201MR174540
  11. [11] I. Kátai and J. Szabó, Canonical number systems for complex integers. Acta Sci. Math. (Szeged) 37 (1975), 255-260. Zbl0309.12001
  12. [12] B. Kovács, Canonical number systems in algebraic number fields. Acta Math. Hungar.37 (1981), 405-407. Zbl0505.12001
  13. [13] B. Kovács and A. Pethö, Number systems in integral domains, especially in orders of algebraic number fields. Acta Sci. Math. (Szeged) 55 (1991), 286-299. Zbl0760.11002
  14. [14] J. Kubilius, Probabilistic methods in the theory of numbers. Amer. Math. Soc. Translations of Math. Monographs, No. 11, Providence, 1964. Zbl0133.30203
  15. [15] C. Mauduit and A. Sárközy, On the arithmetic structure of sets characterized by sum of digits properties. J. Number Theory61 (1996), no. 1, 25-38. Zbl0868.11004
  16. [16] C. Mauduit and A. Sárközy, On the arithmetic structure of the integers whose sum of digits is fixed. Acta Arith.81 (1997), no. 2, 145-173. Zbl0887.11008
  17. [17] W. Narkiewicz, Elementary and analytic theory of algebraic numbers. Springer, Berlin, 1990. Zbl0717.11045
  18. [18] J.M. Thuswaldner, The sum of digits function in number fields: Distribution in residue classes. J. Number Th.74 (1999), 111-125. Zbl0932.11068
  19. [19] ____, The fundamental lemma of Kubilius and the model of Kubilius in number fields. Number Theory, Diophantine, Computational and Algebraic Aspects (Berlin) (K. Györy et. al., ed.), 1998, pp. 489-499. Zbl0914.11051
  20. [20] W.-B. Zhang, Probabilistic number theory in additive semigroups I. Analytic Number Theory, Proceedings of a Conference in Honor of Heini Halberstam (Boston, Basel, Berlin) (B. C. Berndt et. al., ed.), vol. 2, 1996, pp. 839-885. Zbl0861.11048

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