Additive arithmetic functions meet the inclusion-exclusion principle: Asymptotic formulas concerning the GCD and LCM of several integers

We obtain asymptotic formulas for the sums $\sum_{n_1,\ldots,n_k\le x} f((n_1,\ldots,n_k))$ and $ \sum_{n_1,\ldots,n_k\le x} f([n_1,\ldots,n_k])$ involving the gcd and lcm of the integers $n_1,\ldots,n_k$, where $f$ belongs to certain classes of additive arithmetic functions. In particular, we consider the generalized omega function $\Omega_{\ell}(n)= \sum_{p^\nu \mid\mid n} \nu^{\ell}$ investigated by Duncan (1962) and Hassani (2018), and the functions $A(n)=\sum_{p^\nu \mid\mid n} \nu p$, $A^*(n)= \sum_{p \mid n} p$, $B(n)=A(n)-A^*(n)$ studied by Alladi and Erd\H{o}s (1977). As a key auxiliary result we use an inclusion-exclusion-type identity.

It is more difficult to obtain asymptotic formulas for the sums L f,k (x) concerning the lcm of integers. If the function f is multiplicative, then f ([n 1 , . . . , n k ]), and also f ((n 1 , . . . , n k )), are multiplicative functions of k variables. Therefore, the multiple Dirichlet series ∞ n 1 ,...,n k =1 f ([n 1 , . . . , n k ]) n s 1 1 · · · n s k k can be expanded into an Euler product, and the multiple convolution method can be used to deduce asymptotic formulas. For example, the counterpart of (1.2) is n 1 ,...,n k ≤x τ ([n 1 , . . . , where k ≥ 2, Q k (t) is a polynomial in t of degree k and θ is the exponent in the Dirichlet divisor problem. See [19,Th. 3.4]. This approach does not furnish a formula with remainder term, as a counterpart of (1.1). It was only proved in [11,Th. 2.3] that for k ≥ 3, where P 2 k −1 (t) is a polynomial in t of degree 2 k − 1 and r is a positive real number. This conjecture was proved in [6] by a different method, using analytic techniques. See [6,11,18,19] for more details.
In this paper we obtain asymptotic formulas for the sums G f,k (x) and L f,k (x) in the case of certain classes of additive functions. In particular, we consider the generalized omega function Ω ℓ (n) = p ν ||n ν ℓ investigated by Duncan [5] and Hassani [9], and the functions A(n) = p ν ||n νp, A * (n) = p|n p, B(n) = A(n) − A * (n) studied by Alladi and Erdős [1]. A key identity of our approach is the application of the inclusion-exclusion principle to additive functions. See Proposition 1, which may be known in the literature, but we could not find any reference. Other key results used in the proofs are Saffari's estimate obtained for the sum n≤x ω(n) and the estimate for n≤x A(n), where A(n) is the Alladi-Erdős function. See [1,16,17]. The main results on the asymptotic formulas are formulated in Section 3. Some preliminary lemmas needed to the proofs are included in Section 4, and the proofs of the main results are given in Section 5.

Additive functions and the inclusion-exclusion principle
We recall that an arithmetic function f : . Some examples of additive functions are log n, ω(n) and Ω(n).
We generalize these identities to several integers.
Proposition 1. Let f : N → C be an additive function, let k ∈ N and n 1 , . . . , n k ∈ N. Then Proof. It is enough to prove identity (2.1) if n 1 = p ν 1 , . . . , n k = p ν k are powers of the same prime p with ν 1 , . . . , n k ∈ N 0 , that is, By symmetry we can assume that ν 1 ≤ ν 2 ≤ · · · ≤ ν k . Then the LHS of (2. 2) is f (p ν k ). Let 1 ≤ ℓ ≤ k. On the RHS of (2.2), for a fixed j, the term f (p ν ℓ ) appears if i 1 = ℓ < i 2 < · · · < i j ≤ k. This happens k−ℓ j−1 times. Hence on the RHS the coefficient of f (p ν ℓ ) is which completes the proof, similar as in the proof of the inclusion-exclusion principle.
We also recall that a function g : N → C is multiplicative if g(mn) = g(m)g(n) for all m, n ∈ N with (m, n) = 1. If g is multiplicative, then g(n) = p ν ||n g(p ν ). If f is additive, then the function g(n) = 2 f (n) is multiplicative. Conversely, if g is multiplicative (and positive), then the function f (n) = log g(n) is additive. If g is multiplicative, then in a similar manner as above, g((m, n))g([m, n]) = g(m)g(n) holds for every m, n ∈ N. This is well-known and is included in many textbooks. See., e.g., [12,Ex. 1.9]. Also see [10] for the related notion of semimultiplicative (Selberg multiplicative) functions, and [2] for some other similar two variables identities. More generally than (2.3), we have the next result.
Corollary 1. Let g : N → C be a nonvanishing multiplicative function, let k ∈ N and n 1 , . . . , n k ∈ N. Then Proof. Apply (formally) Proposition 1 to the additive function f (n) = log g(n).
3 Asymptotic formulas for multivariable sums

The class F 0 of omega-type functions
Let F 0 denote the class of additive functions f : N → C such that f (p) = 1 for every prime p and f (p ν ) ≪ ν ℓ holds uniformly for the primes p and ν ≥ 2, where ℓ ∈ N 0 is some integer. For example, the functions ω(n) and Ω(n) are in F 0 . More generally, the function Ω ℓ (n) = p ν ||n ν ℓ , with ℓ ∈ N 0 , is in the class F 0 . Note that Ω 0 (n) = ω(n), Ω 1 (n) = Ω(n). The function Ω ℓ (n) was defined by Duncan [5] and an asymptotic formula for n≤x Ω ℓ (n) was obtained in that paper.
Another example of a function in the class F 0 is given by is the number of k-combinations with repetitions of n elements. Observe that T 0 (n) = ω(n), T 1 (n) = Ω(n).
Saffari [16] proved that the estimate holds for every fixed integer N ≥ 1, where M is the Mertens constant defined by and the constants a j (1 ≤ j ≤ N ) are given by By using the proximity of the functions f ∈ F 0 and the function ω we first prove the following result.
for every fixed N ≥ 1, where the constant C f is given by Note that a weaker asymptotic formula for the sum n≤x f (n) is given in [4,Th. 6.19] under the more restrictive conditions f additive, f (p) = 1 and for all primes p and f (p ν ) − f (p ν−1 ) = O(1) uniformly for the primes p and ν ≥ 2, satisfied by the functions ω(n) and Ω(n), but not by Ω ℓ (n) and T ℓ (n) with ℓ ≥ 2.
Corollary 2. The estimate of Theorem 1 holds for the function f (n) = Ω ℓ (n) (ℓ ∈ N 0 ) with the constant where ℓ ν are the Eulerian numbers to be defined in Section 4.2, and the inner sum is considered to be 1 if ℓ = 0.
The result of Corollary 2 was proved by Hassani [9], also by using Saffari's estimate for the sum n≤x ω(n), but invoking some different arguments and without referring to the Eulerian numbers.
Corollary 3. The estimate of Theorem 1 holds for the function f (n) = T ℓ (n) (ℓ ∈ N 0 ) with the constant Next we deduce the following estimates for the sums G f,k (x) with k ≥ 2.
Theorem 2. Let f be a function in class F 0 and let k ∈ N, k ≥ 2. Then where the constant D f,k is given by Corollary 4. The estimate of Theorem 2 holds for the functions f (n) = Ω ℓ (n) and f (n) = T ℓ (n) (ℓ ∈ N 0 ) with the constants where the inner sum is considered to be 1 if ℓ = 0, and In particular, Theorem 2 holds for the functions ω(n) and Ω(n) with the constants In what follows we obtain our estimates for the sums L f,k (x) involving the lcm of integers.
Theorem 3. Let f be a function in class F 0 . Then for every k ∈ N, k ≥ 2, for every fixed N ≥ 1, where the constant E f,k is given by and the constants a j (1 ≤ j ≤ N ) are as in (3.2).
Corollary 5. The estimate of Theorem 3 holds for the functions Ω ℓ (n) and T ℓ (n) with ℓ ∈ N 0 . In particular, it holds for the functions ω(n) and Ω(n) with the constants and Finally, we remark that it is also possible to apply our results to functions f (n) = c log g(n), where g(n) are certain multiplicative functions and c is a constant. For example, let g(n) = τ (n).
Then the function f (n) = log τ (n) log 2 is in the class F 0 , and our results can be applied. We obtain from Theorem 3 the following asymptotic formula.
for every fixed N ≥ 1, where the constant A k is given by In the special case k = 1, the result of Corollary 6 has been obtained by Hassani [8, Th. 1.1], and a weaker asymptotic formula is given in [4, Problem 6.12].

The class F 1 of Alladi-Erdős-type functions
Let F 1 denote the class of additive functions f : N → C such that f (p) = p for every prime p and f (p ν ) ≪ ν ℓ p ν holds uniformly for the primes p and ν ≥ 2, where ℓ ∈ N 0 is some integer. For example, the functions A ℓ (n) = p ν ||n ν ℓ p and A ℓ (n) = p ν ||n ν ℓ p ν with ℓ ∈ N 0 are in F 1 . In particular, A 1 (n) = A(n) := p ν ||n νp and A 0 (n) = A * (n) := p|n p are the Alladi-Erdős functions.
It is known that which can be proved by using a strong form of the prime number theorem. See [1], [17, p. 62, 467]. Also see [20, Th. 2] for a simple approach leading to a slightly weaker error term. The same formula (3.4) holds for n≤x A * (n) = p≤x p x p . We point out the following result. Next we deduce the corresponding estimates for the sums G f,k (x) with k ≥ 2.
Theorem 5. Let f be a function in class F 1 . Then and if k ≥ 3, then Corollary 7. The estimate of Theorem 5 holds for the function f (n) = A ℓ (n) (ℓ ∈ N) with the constants the inner sums being considered 1 if ℓ = 0.
In particular, it holds for the function A(n) = A 1 (n) with and it holds for the function A * (n) = A 0 (n) with the constants D A * ,2 = M the Mertens constant, and D A * ,k = p Now we obtain the estimate for the sums L f,k (x) involving the lcm of integers.
Theorem 6. Let f be a function in class F 1 . Then for every k ∈ N,

The function B(n)
In this section we consider the function B(n) = A(n) − A * (n), where B(p) = 0 for every prime p. It would be possible to define and study here another class of additive functions f with f (p) = 0 for the primes p and with adequate order conditions on f (p ν ) with ν ≥ 2, but we confine ourselves to the function B(n).
Alladi and Erdős [1, Th. 1.5] proved that We improve this estimate in the following way.
For the sums involving the gcd and lcm we have the following results.

Properties of additive functions
The next result is well-known.
Lemma 2. If f is an additive function, then where otherwise.
Hence f = g * 1 and by Möbius inversion we have µ * f = g.
We will use the following identity.

An identity involving the Eulerian numbers
Let n k denote the (classical) Eulerian numbers, defined as the number of permutations h ∈ S n with k descents. Here a number i is called a descent of h if h(i) > h(i + 1). In the paper we use the identity where in the RHS the sum is considered to be 1 if n = 0.
We deduce the following estimates.
Proof. According to identity (4.1), where ℓ is fixed, we have finitely many terms, and the largest term -with respect to p -of the sum is that for n = 0. This gives (4.2). Similarly, ∞ n=2 n ℓ p nk = − where, since ℓ 0 = 1, the term 1/p k cancels out, giving (4.3).

Estimates of certain sums
The estimates of Lemma 4 are not sufficient for our proofs. We need good estimates on the sums n>z n ℓ p nk , where z ≥ 1 is a real number.
Lemma 5. i) Let k, ℓ ∈ N, p be a prime and z 1 be a real number satisfying z > ℓ max(1,log z) Note that (4.2) and (4.3) are special cases of (4.5). The proof of Lemma 5 uses the following van der Corput type bound. Lemma 6. Let a < b be real numbers and g ∈ C 1 [a, b] such that g ≥ 0, g ′ is non-decreasing and there exists λ 1 > 0 such that, for all Proof of Lemma 6. Use the following inequality due to Ostrowski. See, e.g., [13,  Select F (t) = 1/g ′ (t) and G(t) = g ′ (t)e −g(t) and proceeding as in van der Corput's first derivative test, we have as asserted.
Proof of Lemma 5. Apply Lemma 6 with g(t) := tk log p − ℓ log t, for which g ′ (t) = k log p − ℓ t ≥ k log p − ℓ z for all t ≥ z. Note that the condition z > ℓ max(1,log z) k log p ensures that both g(z) > 0 and g ′ (z) > 0. Lemma 6 then yields The second part of the lemma follows by noticing that, if k log p > ℓ z + 1 2 , then 1 k log p−ℓ/z < 2.
5 Proofs of the asymptotic formulas 5.1 Proofs of the results in Section 3.1 Proof of Theorem 1. Using Lemma 3 for k = 1, and that f (p) = 1, again by (4.3) or by (4.5).
Proof of Theorem 9. Similar to the proofs of Theorems 3 and 6, by using Proposition 1 and Theorems 7, 8. We omit the details.