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A characterization of some additive arithmetical functions, III

Jean-Loup Mauclaire (1999)

Acta Arithmetica

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...

Arithmetic properties of positive integers with fixed digit sum.

Florian Luca (2006)

Revista Matemática Iberoamericana

In this paper, we look at various arithmetic properties of the set of those positive integers n whose sum of digits in a fixed base b > 1 is a fixed positive integer s. For example, we prove that such integers can have many prime factors, that they are not very smooth, and that most such integers have a large prime factor dividing the value of their Euler φ function.

Calculating a determinant associated with multiplicative functions

P. Codecá, M. Nair (2002)

Bollettino dell'Unione Matematica Italiana

Let h be a complex valued multiplicative function. For any N ∈ N , we compute the value of the determinant D N := det i | N , j | N ⁡ h ⁢ i , j i ⁢ j where i , j denotes the greatest common divisor of i and j , which appear in increasing order in rows and columns. Precisely we prove that D N = ∏ p l ∥ N 1 p l ⁢ l + 1 ⁢ ∏ i = 1 l h ⁢ p i - h ⁢ p i - 1 τ ⁢ N / p l . This means that D N 1 / τ ⁢ N is a multiplicative function of N . The algebraic apparatus associated with this result allows us to prove the following two results. The first one is the characterization of real multiplicative functions f ⁢ n , with 0 ≤ f ⁢ p < 1 , as minimal values of certain...

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