Note on arithmetical functions and multiples

is a ring of functions with the unity element e(n), wheree(1) = 1 ande(n) = 0 if n > 1. We denote as usual by μ(n) the Möbius function, and by ω(n), (n) the numbers of primes dividingn counted without and with multiplicity. We use the concepts of additive and multiplicative functions in the usual number-theoretic sense. An arbitrary arithmetical functionf can be viewed as a result of convolution of some arithmetical function w and the constant function I (n) = 1: f (n) = (w ∗ I)(n) = ∑

We use this representation as generic and write f (n) = f (n|w). It is our aim to investigate some relations between the conditions set on w and properties of f (n|w).
It is easy to find out which functions w(n) generate additive or multiplicative functions f (n|w).
The function f (n|w) is additive if and only if w(n) = 0 for all n with the condition ω(n) = 1.
Proof. The first statement can be found in most textbooks of number theory. Let us prove the second statement. It is obvious, that the conditions on w imply additivity of f (n|w). We prove that these conditions are necessary. It can be done easily by induction over the values of (n).
and w(n) = 0 follows. Let the statement be true for all n with the condition 2 ω(n) (n) m. Let for some n, ω(n) 2, (n) = m+1. Then n = n p a , where p is prime and (n , p) = 1. We have The last sum is zero and for each δp b , except for the largest δp b = n p a , the condition 2 ω(δp b ) (δp b ) m is satisfied. Hence w(δp b ) = 0, and w(n) = w(n p a ) = 0. The theorem is proved.
For an arbitrary subset of natural numbers A ⊂ N we denote the set of multiples The value of f (n|w), if f (n|w) > 0, can be interpreted as the "weight" of the multiple n in the obvious sense.
We introduce two systems of densities. If A ⊂ N and x > 1, let us denote We denote the lower and the upper limits of ν x {A}, λ x {A}, as x → ∞, by ν{A}, ν{A}, λ{A}, λ{A}, respectively. It is well known that for all subsets A ⊂ N ν{A} λ{A} λ{A} ν{A}.
If ν{A} = ν{A}, we denote this value by ν{A} and say that A possess the numbertheoretic density. If λ{A} = λ{A} = λ{A}, we say that A has the logaritmic density.
We are going to prove some facts about the existence of densities for the sets {n: f (n|w) z}.
then for any z the density ν{n: f (n|w) z} exists.
Proof. Let d 1 < d 2 < . . . be the sequence of all numbers with the property w(d) = 0. Let > 0 and N be some number such that Then Hence, it suffices to show the existence of ν{f (n|w * ) z}. We turn now to the question of existence λ{f (n|w) z}. We use in our reasoning the fact established by Erdös and Davenport: for any subset A ⊂ N the set of multiples M(A) has the logarithmic density (see [3]; [4] Th. 12, p.258; [5] Th. 02, p.5).
here [T ; 2T ) means the set of natural numbers in this interval. The existence of densities in (2) can be proved using the combinatorial including-excluding principle, which works because of finitness of [T ; 2T ). Let k 1 be some natural number. We have, obviously, that It follows then from (2) that Let z 1 = c, z 1 < z 2 < . . . be an arbitrary unbounded sequence. We define a function The existence of λ{f (n|w) z} follows from the previous theorem.
For fixed z c find some z m such that z z m . Obviously, We show that ν{n: f (n|w) z m } 1 − 2 −k and ν{n: f (n|w) c} . The second inequality follows from We obtain the first one using the bound This suffices for the proof.
We are now going to interpret the Behrend inequality for the set of multiples in the context of arithmetical functions. Let A, B be arbitrary subsets of natural numbers. The Behrend inequality is Now from (4) we obtain Evidently, (5) can be rewritten for more than two functions. For which additive or multiplicative functions f 1 (n) = f (n|w 1 ), f 2 (n|w 2 ) inequality (5) holds? Having in mind Theorem 1, we derive quickly the sufficient condition: it suffices that for any prime p 0 f i (p) f i (p 2 ) . . . , i = 1, 2, . . . , holds. If the sets {n: f i (n) < z i }, {n: f 1 (n) < z 1 , f 2 (n) < z 2 } possess the numbertheoretic densities they can be used in (5) instead of logaritmic ones. For example, let P 1 , P 2 be some arbitrary subsets of prime numbers; define the additive functions f i (n) = p∈P i ,p α ||n (α − 1), i = 1, 2.