PL EN

Preferencje
Język
Widoczny [Schowaj] Abstrakt
Liczba wyników
Czasopismo

## Acta Arithmetica

1999 | 91 | 3 | 229-232
Tytuł artykułu

### A characterization of some additive arithmetical functions, III

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
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.
Słowa kluczowe
Czasopismo
Rocznik
Tom
Numer
Strony
229-232
Opis fizyczny
Daty
wydano
1999
otrzymano
1998-01-05
poprawiono
1998-12-11
Twórcy
autor
• Institut de Mathématiques, U.M.R. 7586 du C.N.R.S., Case 247, 4 Place Jussieu, 75252 Paris Cedex 05, France
Bibliografia
• [1] P. D. T. A. Elliott, The value distribution of differences of additive arithmetic functions, J. Number Theory 32 (1989), 339-370.
• [2] P. Erdős, On the distribution function of additive functions, Ann. of Math. 47 (1946), 1-20.
• [3] I. Kátai, Some results and problems in the theory of additive functions, Acta Sci. Math. (Szeged) 30 (1969), 305-311.
• [4] I. Kátai, On number theoretical functions, in: Number Theory (Colloq., János Bolyai Math. Soc., Debrecen, 1968), North-Holland, Amsterdam, 1970, 133-137.
• [5] J. L. Mauclaire, Contributions à la théorie des fonctions additives, thèse, Publ. Math. Orsay 215, 1977.
• [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] N. M. Timofeev, Integral limit theorems for sums of additive functions with shifted arguments, Izv. Math. 59 (1995), 401-426.
Typ dokumentu
Bibliografia
Identyfikatory