A characterization of some additive arithmetical functions, III

Jean-Loup Mauclaire

Acta Arithmetica (1999)

  • Volume: 91, Issue: 3, page 229-232
  • ISSN: 0065-1036

Abstract

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I. Introduction. In 1946, P. Erdős [2] proved that if a real-valued additive arithmetical function f satisfies the condition: f(n+1) - f(n) → 0, n → ∞, then there exists a constant C such that f(n) = C log n for all n in ℕ*. Later, I. Kátai [3,4] was led to conjecture that it was possible to determine additive arithmetical functions f and g satisfying the condition: there exist a real number l, a, c in ℕ*, and integers b, d such that f(an+b) - g(cn+d) → l, n → ∞. This problem has been treated essentially by analytic methods ([1], [7]). In this article, we shall provide, in an elementary way, a characterization of real-valued additive arithmetical functions f and g satisfying the condition: (H) there exist a and b in ℕ* with (a,b) = 1 and a finite set Ω such that (1) lim_{n→∞} min_{ω∈Ω} |f(an+b) - g(n) - ω| = 0.

How to cite

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Jean-Loup Mauclaire. "A characterization of some additive arithmetical functions, III." Acta Arithmetica 91.3 (1999): 229-232. <http://eudml.org/doc/207353>.

@article{Jean1999,
abstract = { I. Introduction. In 1946, P. Erdős [2] proved that if a real-valued additive arithmetical function f satisfies the condition: f(n+1) - f(n) → 0, n → ∞, then there exists a constant C such that f(n) = C log n for all n in ℕ*. Later, I. Kátai [3,4] was led to conjecture that it was possible to determine additive arithmetical functions f and g satisfying the condition: there exist a real number l, a, c in ℕ*, and integers b, d such that f(an+b) - g(cn+d) → l, n → ∞. This problem has been treated essentially by analytic methods ([1], [7]). In this article, we shall provide, in an elementary way, a characterization of real-valued additive arithmetical functions f and g satisfying the condition: (H) there exist a and b in ℕ* with (a,b) = 1 and a finite set Ω such that (1) lim\_\{n→∞\} min\_\{ω∈Ω\} |f(an+b) - g(n) - ω| = 0. },
author = {Jean-Loup Mauclaire},
journal = {Acta Arithmetica},
keywords = {real-valued additive arithmetical functions},
language = {eng},
number = {3},
pages = {229-232},
title = {A characterization of some additive arithmetical functions, III},
url = {http://eudml.org/doc/207353},
volume = {91},
year = {1999},
}

TY - JOUR
AU - Jean-Loup Mauclaire
TI - A characterization of some additive arithmetical functions, III
JO - Acta Arithmetica
PY - 1999
VL - 91
IS - 3
SP - 229
EP - 232
AB - I. Introduction. In 1946, P. Erdős [2] proved that if a real-valued additive arithmetical function f satisfies the condition: f(n+1) - f(n) → 0, n → ∞, then there exists a constant C such that f(n) = C log n for all n in ℕ*. Later, I. Kátai [3,4] was led to conjecture that it was possible to determine additive arithmetical functions f and g satisfying the condition: there exist a real number l, a, c in ℕ*, and integers b, d such that f(an+b) - g(cn+d) → l, n → ∞. This problem has been treated essentially by analytic methods ([1], [7]). In this article, we shall provide, in an elementary way, a characterization of real-valued additive arithmetical functions f and g satisfying the condition: (H) there exist a and b in ℕ* with (a,b) = 1 and a finite set Ω such that (1) lim_{n→∞} min_{ω∈Ω} |f(an+b) - g(n) - ω| = 0.
LA - eng
KW - real-valued additive arithmetical functions
UR - http://eudml.org/doc/207353
ER -

References

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  1. [1] P. D. T. A. Elliott, The value distribution of differences of additive arithmetic functions, J. Number Theory 32 (1989), 339-370. Zbl0707.11058
  2. [2] P. Erdős, On the distribution function of additive functions, Ann. of Math. 47 (1946), 1-20. Zbl0061.07902
  3. [3] I. Kátai, Some results and problems in the theory of additive functions, Acta Sci. Math. (Szeged) 30 (1969), 305-311. Zbl0186.35901
  4. [4] I. Kátai, On number theoretical functions, in: Number Theory (Colloq., János Bolyai Math. Soc., Debrecen, 1968), North-Holland, Amsterdam, 1970, 133-137. 
  5. [5] J. L. Mauclaire, Contributions à la théorie des fonctions additives, thèse, Publ. Math. Orsay 215, 1977. 
  6. [6] J. L. Mauclaire, Sur la régularité des fonctions additives, Sém. Delange-Pisot-Poitou (15ième année, 1973/1974), fasc. 1, exp. no. 23, 1975. 
  7. [7] N. M. Timofeev, Integral limit theorems for sums of additive functions with shifted arguments, Izv. Math. 59 (1995), 401-426. Zbl0894.11034

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