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.
@article{bwmeta1.element.bwnjournal-article-aav91i3p229bwm,
author = {Jean-Loup Mauclaire},
title = {A characterization of some additive arithmetical functions, III},
journal = {Acta Arithmetica},
volume = {89},
year = {1999},
pages = {229-232},
zbl = {0969.11034},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-aav91i3p229bwm}
}
Jean-Loup Mauclaire. A characterization of some additive arithmetical functions, III. Acta Arithmetica, Tome 89 (1999) pp. 229-232. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-aav91i3p229bwm/
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