Functions concerned with divisors of order $r$

N. Minculete has introduced a concept of divisors of order $r$: integer $d=p_1^{b_1}\cdots p_k^{b_k} $ is called a divisor of order $r$ of $n=p_1^{a_1}\cdots p_k^{a_k}$ if $d \mid n$ and $b_j\in\{r, a_j\}$ for $j=1,\ldots,k$. One can consider respective divisor function $\tau^{(r)}$ and sum-of-divisors function $\sigma^{(r)}$. In the present paper we investigate the asymptotic behaviour of $\sum_{n\le x} \tau^{(r)}(n)$ and $\sum_{n\le x} \sigma^{(r)}(n)$. We also provide conditional estimates under Riemann hypothesis.


Introduction
Recently N. Minculete in his PhD Thesis [10], devoted to the functions using exponential divisors, and in further paper [11] introduced a concept of divisors of order r: integer d = p b1 1 · · · p b k k is called a divisor of order r of number n = p a1 1 · · · p b k k if d divides n in the usual sense and b j ∈ {r, a j } for j = 1, . . . , k. We also suppose that 1 is a divisor of any order of itself (but not of any other number). Let us denote respective divisor and sum-of-divisor functions as τ (r) and σ (r) . These functions are multiplicative and τ (r) (p a ) = 1, a r, 2, a > r. (1) σ (r) (p a ) = p a , a r, p a + p r , a > r. (2) In a special case of r = 0 we get well-studied unitary divisors. For example, it was proved in [3] that (under Riemann hypothesis error term is O(x 221/608+ε ) due to [7]) and in [14] it was proved that In another special case of r = 1 we get so-called by Minculete exponential semiproper divisors and denote τ (e)s := τ (1) , σ (e)s := σ (1) . An integer d is an exponential semiproper divisor of n if ker d = ker n and (d/ ker n, n/d) = 1, where ker n = p|n p.
Minculete proved in [10, (3.1.17-19)] that lim sup n→∞ log τ (r) (n) log log n log n = log 2 r + 1 , In the present paper we improve the error term in (6) and establish asymptotic formulas for n x σ (r) (n) with O-and Ω-estimates of the error term.

Notation
In asymptotic relations we use ∼, ≍, Landau symbols O and o, big omegas Ω and Ω ± , Vinogradov symbols ≪ and ≫ in their usual meanings. All asymptotic relations are given as an argument tends to the infinity.
Letter p with or without indexes denote rational prime. As usual ζ(s) is Riemann zeta-function. For complex s we denote σ := ℜs and t := ℑs.
We use abbreviations llog x := log log x, lllog x := log log log x.
Letter γ denotes Euler-Mascheroni constant, γ ≈ 0.577. Everywhere ε > 0 is an arbitrarily small number (not always the same even in one equation).
We write f ⋆ g for Dirichlet convolution: Function ker : N → N stands for ker n = p|n p. For a set A notation #A means the cardinality of A.
In fact ∆(a, b; x) can be estimated more precisely. For our goals we are primarily interested in the behaviour of ∆(1, b; x). Let us suppose that then due to [8,Th. 5.11] we can choose Estimates for b 16 are given in Table 1. Estimate for b = 1 belongs to Huxley [5], and estimate for b = 2 belongs to Graham and Kolesnik [4]. We have found no references on the best known results for b 3, so we calculated them with the use of [8, Th. 5.11, Th. 5.12] selecting appropriate exponent pairs carefully. It seems that some of this estimates may be new.
So µ k (n k ) = µ(n) and µ k (m) = 0 for all other arguments. Trivially µ 1 ≡ µ. Then [6,Th. 12.7] for the proof of the last estimate. Assuming Riemann hypothesis (RH) we get much better result Proof. This is a simplified version of [9, Th. 2].
4. Asymptotic properties of τ (r) (n) Lemma 4. Let F r (s) be Dirichlet series for τ (r) : Proof. Let us transform Bell series for τ (r) : The representation of F r in the form of an infinite product by p completes the proof: It follows from (9) that (10) τ (r) = τ (1, r + 1; ·) ⋆ µ 2r+2 where constants A and B are specified below in (11).
Proof. Taking into account (10) we have for r > 0 n x For the case r = 0 see (3) above.
First of all consider and S 2 is the rest of n x τ (r−1) (n). We note that under RH by taking into account y x 1/2r we have and so
Then by Perron formula with c = 1 + ε, T = x 2 one can estimate
Theorem 2. If ∆ is estimated as in (8) and θ r < 1/2r then under RH Proof. Let us start with (12): accomplishes the proof.
For the values of θ b from Table 1 we have So currently the only non-trivial case of the previous theorem is an estimation for τ (1) ≡ τ (e)s . We get under assumption of RH that Proof. Equation (14) is implied by the substitution m 1 = 1, m 2 = r, n 1 = 2r into Lemma 3. The choice of parameters plainly follows from (9). We obtain which is an exponent in the required Ω-term.
Proof. Consider Bell series for σ (r) : and For σ > 1 we have Now (15) follows from the representation of G r in the form of infinite product by p: Following theorem generalizes (4).
Proof. For a fixed r let z(n) be the coefficient at n −s of the Dirichlet series ζ(s − 1)ζ (r + 1)s − r ζ (r + 2)s − r − 1 and let h(n) be the coefficient of the Dirichlet series H r (s). It follows from (15) that σ (r) = z ⋆ h. One can verify that z(n) = ab r+1 c r+2 =n ab r c r+1 µ(c).

Some remarks
The estimate (5) implies that τ (r) (n)/n → 0 as n → ∞. Thus it is natural to ask what is the maximum value of this ratio. Proof. Recalling the definition (1) we obtain that the least value of a for which τ (r) (p a ) is different from 1 is a = r + 1. So τ (r) (n) = 2 #{p r+1 |n} 2 (log 2 n)/(r+1) = n 1/(r+1) and the statement of the lemma easily follows.
Then S N (q, r; λ) weakly converges to a function S(q, r; λ) which is continuous if and only if q > r.
Proof. Let us fix arbitrary q and r and let f (n) := ln σ (r) (n q ) n q , here f is an additive function. It is enough to prove that F N (λ) := 1 N #{n N | f (n) λ} converges weakly to some F (λ) as N → ∞ and F is continuous if and only if q > r. By definition (2) σ (r) (p q ) = p q , r q, p q + p r , r < q.