Generalized Alternating Sums of Multiplicative Arithmetic Functions

Rimer Zurita Carrera de Matemática Universidad Mayor de San Andrés Avenida Villazón 1995 Planta Baja del Edificio Viejo Monoblock Predio Central La Paz Bolivia

                 rzuritao@fcpn.edu.bo

Abstract

In this paper, given a finite set of primes Q, we derive asymptotic formulas for generalized alternating sums of the form ∑_{n≤x} t_{Q}(n)f(n) and ∑_{n≤x} t_{Q}(n)\frac{1}{f(n)}, where f is a multiplicative arithmetic function, and t_{Q}(n) equals −1 if n is divisible by some prime q ∈ Q, and 1 otherwise. In particular, these results are applicable to known functions, such as Euler’s totient function, the sum of divisors function, the divisor function, and others. In the particular case of Q = {2}, we generalize various results obtained by Tóth, even improving one of his results proposed as open problem.

1 Introduction

Throughout this paper, we let ℙ denote the set of all prime numbers, and Q = {q_{1}, q_{2}, …, q_{r}} be any finite set of prime numbers. We use the notation q_{min} := min{q_{i}} and q_{max} := max{q_{i}}. The letter p will always stand for a prime number.

Alternating sums appear in various topics of mathematics, including number theory. For example, Bordellès and Cloitre [1] established asymptotic formulas with error terms for alternating sums of the form

∑_{n≤x}(−1)^{n−1} \frac{1}{g(n)},

where g belongs to a class of multiplicative functions, including Euler’s totient function φ, the sum-of-divisors function σ and the Dedekind function ψ.

Tóth [11] established some general results for alternating sums of the form

∑_{n≤x}(−1)^{n−1}f(n) or ∑_{n≤x}(−1)^{n−1} \frac{1}{f(n)},

where f belongs to a broader class of multiplicative arithmetic functions than those considered by Bordellès and Cloitre [1], extending their results to a whole new kind of multiplicative functions, such as the divisor function τ, the gcd-sum function P, the square free kernel κ, the square free numbers μ^{2} function, the number of abelian groups a(n), the sum-of-unitary-divisor function σ^{∗}, the unitary-Euler function φ^{∗}, the unitary-squarefree kernel κ^{∗}, the powerful part of a number, and the sum-of-bi-unitary-divisors function.

In the last part of his paper, Tóth proposes a generalization for alternating sums, and also finds an asymptotic result for the generalized sum

∑_{n≤x} t_{Q}(n)σ(n),

where

t_{Q}(n) = \begin{cases}1, & if q ∤ n for all q ∈ Q; \\ −1, & otherwise, \end{cases}

is defined for a finite set of prime numbers Q, and σ(n) = ∑_{d|n} d is the sum of divisors function.

Let

D_{Q}(f, s) := ∑_{n=1}^{∞} t_{Q}(n)\frac{f(n)}{n^{s}},

be the Dirichlet series for the multiplicative function f, with generalized alternating signs depending on Q. For example, if Q = {2, 3}, we have

D_{\{2,3\}}(f, s) = \frac{f(1)}{1^{s}} − \frac{f(2)}{2^{s}} − \frac{f(3)}{3^{s}} − \frac{f(4)}{4^{s}} + \frac{f(5)}{5^{s}} − \frac{f(6)}{6^{s}} + \frac{f(7)}{7^{s}} + ⋯

Tóth [11, Prop. 56] proved that

D_{Q}(f, s) = D(f, s) (2 ∏_{q∈Q} (∑_{v=0}^{∞} \frac{f(q^{v})}{q^{vs}})^{−1} − 1) ,  (1)

where D(f, s) is the Dirichlet series associated with the multiplicative arithmetic function f.

For q ∈ Q, let us consider the formal power series

S_{f,q}(x) := 1 + ∑_{v=1}^{∞} f(q^{v})x^{v},

and its inverse formal series

\overline{S}_{f,q}(x) := 1 + ∑_{v=1}^{∞} b_{v,q}x^{v}.

From (1) we have, by convolution, that

∑_{n≤x} t_{Q}(n)f(n) = ∑_{d≤x} h_{f,Q}(d) ∑_{j≤x/d} f(j),

where

h_{f,Q}(n) = \begin{cases}2b_{v_{1},q_{1}} ⋯ b_{v_{r},q_{r}}, & if n = q_{1}^{v_{1}} ⋯ q_{r}^{v_{r}}; \\ 1, & if n = 1; \\ 0, & otherwise. \end{cases}

From this expression we can derive asymptotic formulas for

∑_{n≤x} t_{Q}(n)f(n),  (2)

as long as asymptotic formulas are known for ∑_{n≤x} f(n), and the coefficients b_{v,q} are adequately estimated.

For example, Tóth [11, Teo. 57] proved that

∑_{n≤x} t_{Q}(n)σ(n) = \frac{π^{2}}{12} (2 ∏_{q∈Q} (1 − \frac{1}{q}) (1 − \frac{1}{q^{2}} ) − 1) x^{2} + O(x(log x)^{2/3}).  (3)

In this paper, we obtain asymptotic expressions for (2), among whose applications we obtain the result (3) and others.

2 Main results

Theorem 1. Let f be a multiplicative function and consider the following four conditions

(i) there exists a constant C_{f} such that

∑_{n≤x} f(n) = C_{f}x^{2} + O(xR_{f}(x)),

where 1 ≪ R_{f}(x) when x → ∞ and R_{f}(x) is an increasing function; (ii) S_{f,q_{i}} (\frac{1}{q_{i}^{2}} ) converges for all i; (iii) the sequence (b_{v,q_{i}})_{v≥0} satisfies b_{v,q_{i}} ≪ 1 for all i; (iv) the sequence (b_{v,q_{i}})_{v≥0} satisfies |b_{v,q_{i}}| ≪ (r_{i})^{v} with 1 ≤ r_{i} ≤ \frac{q_{i}^{2}}{q_{max}} for all i. Assume that conditions (i) and (ii) hold and that one of the two conditions (iii) or (iv) also holds. Then

∑_{n≤x} t_{Q}(n)f(n) = C_{f}x^{2} (\frac{2}{S_{f,q_{1}}(1/q_{1}^{2}) ⋯ S_{f,q_{n}}(1/q_{n}^{2})} − 1) + O(xR_{f}(x)).

Theorem 2. Let f be a multiplicative function, and let us suppose that (i) there exist constants D_{f} and E_{f} such that

∑_{n≤x} \frac{1}{f(n)} = D_{f}(log x + E_{f}) + O(x^{−1}R_{1/f}(x)),

where 1 ≪ R_{1/f}(x) = o(x) if x → ∞ and R_{1/f}(x) is an increasing function; (ii) the radius of convergence of S_{1/f,q_{i}}(x) is r_{1/f,q_{i}} > 1, for all i; (iii) the coefficients of b_{v,q_{i}} satisfy b_{v,q_{i}} ≪ M_{i}^{v} if v → +∞, for all i and where M_{i} < \frac{q_{min}}{q_{i}}. Then

∑_{n≤x} t_{Q}(n)\frac{1}{f(n)}

= D_{f} ((\frac{2}{∏_{i} S_{1/f,q_{i}}(1)} − 1) (log x + E_{f}) + \frac{2}{∏_{i} S_{1/f,q_{i}}(1)} · ∑_{i=1}^{r} \frac{log(q_{i})S′_{1/f,q_{i}}(1)}{S_{1/f,q_{i}}(1)}) + O(T_{1/f,Q}(x)),

where

T_{1/f,Q}(x) =

x^{−1}R_{1/f}(x),  if max(q_{i}M_{i}) < 1;

x^{−1}R_{1/f}(x)(log x)^{r},  if max(q_{i}M_{i}) = 1;

(log x)^{r−1} max{log x · x^{log M_{max}/ log q_{min}}, x^{log(max(q_{i}M_{i}))/ log q_{min}} · x^{−1}R_{1/f}(x)}, if max(q_{i}M_{i}) > 1.

3 Proofs of the main results

3.1 Proof of Theorem 1

Proof. Under the hypothesis of the theorem, we have that

∑_{n≤x} t_{Q}(n)f(n) = ∑_{d≤x} h_{f,Q}(d) ∑_{j≤x/d} f(j) = ∑_{d≤x} h_{f,Q}(d) (C_{f} \frac{x^{2}}{d^{2}} + O (\frac{x}{d}R_{f}(x/d)))

= C_{f}x^{2} · ∑_{d≤x} \frac{h_{f,Q}(d)}{d^{2}} + O (xR_{f}(x) ∑_{d≤x} \frac{|h_{f,Q}(d)|}{d}) .

On one hand, for some δ′ < 1,

∑_{d≤x} \frac{|h_{f,Q}(d)|}{d} ≤ ∑_{q_{1}^{v_{1}}···q_{r}^{vr} ≤x} \frac{2|b_{v_{1},q_{1}} ⋯ b_{v_{r},q_{r}}|}{q_{1}^{v_{1}} ⋯ q_{r}^{v_{r}}} + 1

≪ ∑_{q_{1}^{v_{1}}···q_{r}^{vr} ≤x} \frac{r_{1}^{v_{1}} ⋯ r_{r}^{v_{r}}}{q_{1}^{v_{1}} ⋯ q_{r}^{v_{r}}} ≪ ∑_{v_{1}+···+v_{r}≤ \frac{log x}{log q_{min}}} (δ′)^{v_{1}+···+v_{r}}

≪ ∑_{n≤ \frac{log x}{log q_{min}}} (δ′)^{n}\binom{n + r − 1}{r − 1} = ∑_{n≤ \frac{log x}{log q_{min}}} (δ′)^{n} \frac{(n + r − 1) ⋯ (n + 2) · (n + 1)}{(r − 1)!}

≪ ∑_{n≤ \frac{log x}{log q_{min}}} (δ′)^{n}(n + r − 1)^{r−1} ≪ 1.

On the other hand, setting s = 2 in (1),

∑_{d≤x} \frac{|h_{f,Q}(d)|}{d^{2}} = ∑_{d=1}^{∞} \frac{|h_{f,Q}(d)|}{d^{2}} − ∑_{d>x} \frac{|h_{f,Q}(d)|}{d^{2}} = \frac{2}{S_{f,q_{1}}(1/q_{1}^{2}) ⋯ S_{f,q_{r}}(1/q_{r}^{2})} − 1,

and

x^{2} ∑_{d>x} \frac{|h_{f,Q}(d)|}{d^{2}} ≪ x^{2} ∑_{q_{1}^{v_{1}}···q_{r}^{vr} >x} \frac{b_{v_{1},q_{1}} ⋯ b_{v_{r},q_{r}}}{q_{1}^{2v_{1}} ⋯ q_{r}^{2v_{r}}} .

Case (iii) of Theorem 1:

x^{2} ∑_{d>x} \frac{|h_{f,Q}(d)|}{d^{2}} ≪ x^{2} ∑_{d>x} \frac{1}{d^{2}} ≪ x ≪ xR_{f}(x).

Case (iv) of Theorem 1:

Let us define δ := max_{i} (\frac{r_{i}}{q_{i}^{2}} ) < \frac{1}{q_{max}} . Then

x^{2} ∑_{d>x} \frac{|h_{f,Q}(d)|}{d^{2}} ≪ x^{2} ∑_{v_{1}+···+v_{r}> \frac{log x}{log qmax}} (\frac{r_{1}}{q_{1}^{2}} )^{v_{1}}⋯ (\frac{r_{r}}{q_{r}^{2}} )^{v_{r}} ≪ x^{2} ∑_{v_{1}+···+v_{r}> \frac{log x}{log qmax}} (δ)^{v_{1}+···+v_{r}}

≪ x^{2} ∑_{n> \frac{log x}{log qmax}} δ^{n}n^{r−1} ≪ x^{2}δ^{\frac{log x}{log qmax}} (log x)^{r−1} = x^{2+ \frac{log δ}{log qmax}} (log x)^{r−1} ≪ xR_{f}(x).

3.2 Proof of Theorem 2

Proof. From the hypothesis of the theorem, we have that

∑_{n≤x} t_{Q}(n)\frac{1}{f(n)} = ∑_{d≤x} h_{1/f,Q}(d) ∑_{j≤x/d} \frac{1}{f(j)}

= ∑_{d≤x} h_{1/f,Q}(d) (D_{f} (log \frac{x}{d} + E_{f}) + O ((\frac{x}{d})^{−1} R_{1/f}(x/d)))

= D_{f}(log x + E_{f}) ∑_{d≤x} h_{1/f,Q}(d) − D_{f} ∑_{d≤x} h_{1/f,Q}(d) log d + O (x^{−1}R_{1/f}(x) · ∑_{d≤x} d|h_{1/f,Q}(d)|)

= D_{f}(log x + E_{f}) ∑_{d=1}^{∞} h_{1/f,Q}(d) + O (log x ∑_{d>x} |h_{1/f,Q}(d)|)

− D_{f} ∑_{d=1}^{∞} h_{1/f,Q}(d) log d + O (∑_{d>x} |h_{1/f,Q}(d)| log d) + O (x^{−1}R_{1/f}(x) ∑_{d≤x} d · |h_{1/f,Q}(d)|) .

In particular, by (1),

∑_{d=1}^{∞} \frac{h_{1/f,Q}(d)}{d^{s}} = \frac{2}{S_{1/f,q_{1}}(1/q_{1}^{s}) ⋯ S_{1/f,q_{r}}(1/q_{r}^{s})} − 1,

then we see that

∑_{d=1}^{∞} h_{1/f,Q}(d) = \frac{2}{S_{1/f,q_{1}}(1) ⋯ S_{1/f,q_{r}}(1)} − 1,

and

∑_{d=1}^{∞} h_{1/f,Q}(d) log d = −2 · (\frac{log q_{1} · S′_{1/f,q_{1}}(1)}{S_{1/f,q_{1}}^{2}(1) ⋯ S_{1/f,q_{r}}(1)} + ⋯ + \frac{log q_{r} · S′_{1/f,q_{r}}(1)}{S_{1/f,q_{1}}(1) ⋯ S_{1/f,q_{r}}^{2}(1)}) .

We also have that

∑_{d>x} |h_{1/f,Q}(d)| ≤ 1 + ∑_{q_{1}^{v_{1}}···q_{r}^{vr} >x} 2|b_{v_{1},q_{1}} ⋯ b_{v_{r},q_{r}}| ≪ ∑_{q_{1}^{v_{1}}···q_{r}^{vr} >x} M_{1}^{v_{1}} ⋯ M_{r}^{v_{r}} ≪ ∑_{q_{1}^{v_{1}}···q_{r}^{vr} >x} M_{max}^{v_{1}+···+v_{r}}

≪ ∑_{v_{1}+···+v_{r}> \frac{log x}{log qmax}} M_{max}^{v_{1}+···+v_{r}} ≪ ∑_{n> \frac{log x}{log qmax}} M_{max}^{n}(n + r − 1)^{r−1} ≪ (\frac{log x}{log q_{max}} )^{r−1} Mmax^{\frac{log x}{log qmax}}

≪ (log x)^{r−1}x^{\frac{log Mmax}{log qmax}} . Similarly,

∑_{d>x} |h_{1/f,Q}(d)| log d ≪ ∑_{q_{1}^{v_{1}}···q_{r}^{vr} >x} |b_{v_{1},q_{1}} ⋯ b_{v_{r},q_{r}}|(v_{1} log q_{1} + ⋯ + v_{r} log q_{r})

≪ ∑_{q_{1}^{v_{1}}···q_{r}^{vr} >x}(M_{max})^{v_{1}+···+v_{r}} log q_{max} · (v_{1} + ⋯ + v_{r}) ≪ ∑_{n> \frac{log x}{log qmax}} (M_{max})^{n}(n + r − 1)^{r−1} · n

≪ ∑_{n> \frac{log x}{log qmax}} (M_{max})^{n}n^{r} ≪ (log x)^{r}x^{log M_{max}/ log q_{max}}.

On the other hand,

∑_{d≤x} d|h_{1/f,Q}(d)| ≪ ∑_{q_{1}^{v_{1}}···q_{r}^{vr} ≤x} q_{1}^{v_{1}} ⋯ q_{r}^{v_{r}}M_{1}^{v_{1}} ⋯ M_{r}^{v_{r}} = ∑_{q_{1}^{v_{1}}···q_{r}^{vr} ≤x}(q_{1}M_{1})^{v_{1}} ⋯ (q_{r}M_{r})^{v_{r}}

≪ ∑_{v_{1}+···+v_{r}≤ \frac{log x}{log q_{min}}} (max(q_{i}M_{i}))^{v_{1}+···+v_{r}} ≪ ∑_{n≤ \frac{log x}{log q_{min}}} (max(q_{i}M_{i}))^{n} · n^{r−1}

≪ \begin{cases}1, & if max(q_{i}M_{i}) < 1; \\ (log x)^{r}, & if max(q_{i}M_{i}) = 1; \\ x^{log max(q_{i}M_{i})/ log q_{min}} · (log x)^{r−1}, & if max(q_{i}M_{i}) > 1. \end{cases}

Case 1: If max(q_{i}M_{i}) < 1, then

(log x)^{r} · x^{log M_{max}/ log q_{max}} ≪ x^{−1}R_{1/f}(x)
⇔ (log x)^{r}x^{1+log M_{max}/ log q_{max}} ≪ R_{1/f}(x),
since q_{max}M_{max} < 1, which implies that log q_{max} + log M_{max} < 0.

Case 2: If max(q_{i}M_{i}) = 1, then (log x)^{r} · x^{log M_{max}/ log q_{max}} ≪ (log x)^{r}x^{−1}R_{1/f}(x)

⇔ x^{1+log M_{max}/ log q_{max}} ≪ R_{1/f}(x).

Case 3: If max(q_{i}M_{i}) > 1,

(log x)^{r}x^{log M_{max}/ log q_{max}} ≪ x^{−1}R_{1/f}(x)x^{\frac{log max(q_{i}M_{i})}{log q_{min}}} (log x)^{r−1}
⇔ (log x) · x^{1+ \frac{log Mmax}{log qmax} − \frac{log max(q_{i}M_{i})}{log q_{min}}} ≪ R_{1/f}(x).

Since all three cases have been considered, the proof of Theorem 2 is complete.

4 Applications

For the applications to various multiplicative functions shown in this section, it is sufficient to verify that conditions (i), (ii), (iii) and (iv) of Theorems 1 and 2 hold.

4.1 Euler’s totient function φ(n)

Let us consider Euler’s totient function φ(n) = n ∏_{p|n} (1 − \frac{1}{p}).

Theorem 3.

∑_{n≤x} t_{Q}(n)φ(n) = \frac{3}{π^{2}} x^{2} (\frac{2}{∏_{q∈Q} (1 + \frac{1}{q})} − 1) + O (x(log x)^{2/3}(log log x)^{4/3}) ,  (4)

and

\begin{matrix}∑_{n≤x} t_{Q}(n)\frac{1}{φ(n)} = A((2 ∏_{q∈Q} \frac{(q − 1)^{2}}{q^{2} − q + 1} − 1)(log x + γ − B) \\ + 2 ∏_{q∈Q} \frac{(q − 1)^{2}}{q^{2} − q + 1} · ∑_{q∈Q} \frac{q^{2} log q}{(q − 1)(q^{2} − q + 1)}) + O(R_{1/φ}(x)), \end{matrix}  (5)

where R_{1/φ}(x) = x^{−1}(log x)^{2/3} if 2 ∉ Q, and R_{1/φ}(x) = x^{−1}(log x)^{2/3+r} if 2 ∈ Q, γ is Euler’s constant, and constants A and B are defined by

A = \frac{ζ(2)ζ(3)}{ζ(6)}, B = ∑_{p∈ℙ} \frac{log p}{p^{2} − p + 1}.

Proof. Concerning (4), we know from Walfisz [12, p. 144] that

∑_{n≤x} φ(n) = \frac{3}{π^{2}} x^{2} + O(x(log x)^{2/3}(log log x)^{4/3}).

Then we have R_{φ}(x) = (log x)^{2/3}(log log x)^{4/3}, so condition (i) of Theorem 1 is satisfied. On the other hand, we can see that

S_{f,q}(x) = 1 + ∑_{v=1}^{∞} φ(q^{v})x^{v} = 1 + ∑_{v=1}^{∞}(q^{v} − q^{v−1})x^{v}

= 1 + (1 − \frac{1}{q}) (\frac{qx}{1 − qx}) = \frac{1 − x}{1 − qx}, |x| < 1/q,

and condition (ii) of Theorem 1 is satisfied.

Then we conclude that

\overline{S}_{φ,q}(x) = \frac{1 − qx}{1 − x} = 1 + \frac{(1 − q)x}{(1 − x)} = 1 + (1 − q)x ∑_{v=0}^{∞} x^{v} (|x| < 1),

so b_{v,q} = (1 − q) ≪ 1, and condition (iii) of Theorem 1 is satisfied. Furthermore, we see that S_{φ,q}(1/q^{2}) = \frac{1−1/q^{2}}{1−q·1/q^{2}} = \frac{q+1}{q}.

Concerning (5), we know from Landau [2, Thm. 1.1] and Sitaramachandraro [4] that

∑_{n≤x} \frac{1}{φ(n)} = A(log x + γ − B) + O(x^{−1}(log x)^{2/3}).

Then we have that R_{1/φ}(x) = (log x)^{2/3}, impliying that condition (i) of Theorem 2 is satisfied. On the other hand, we see that

S_{1/φ,q}(x) = 1 + ∑_{v=1}^{∞} \frac{1}{φ(q^{v})}x^{v} = 1 + ∑_{v=1}^{∞} \frac{x^{v}}{(q^{v} − q^{v−1})}

= 1 + \frac{q}{(q − 1)} \frac{x}{(q − x)} = \frac{x + q(q − 1)}{(q − 1)(q − x)} (|x| < q),

so condition (ii) of Theorem 2 is satisfied.

Finally,

\overline{S}_{1/φ,q}(x) = (q − 1)( − 1 + \frac{q^{2}}{x + q(q − 1)}) = (q − 1)( − 1 + \frac{q}{q − 1} ∑_{v=0}^{∞} \frac{(−1)^{v}}{q^{v}(q − 1)^{v}} x^{v})

(|x| < q(q − 1)),

so b_{v,q} = \frac{(−1)^{v}q}{q^{v}(q−1)^{v}(q−1)} ≪ (\frac{1}{q(q−1)})^{v}, having that M_{q} = \frac{1}{q(q−1)} < \frac{q_{min}}{q}, and condition (iii) of Theorem 2 is satisfied with max(M_{i}q_{i}) = 1/(q_{min} − 1) = 1 if q_{min} = 2, and max(M_{i}q_{i}) < 1 if q_{min} > 2. Furthermore, we see that

S_{1/φ,q}(1) = \frac{q^{2} − q + 1}{(q − 1)^{2}} and S′_{1/φ,q}(1) = \frac{q^{2}}{(q − 1)^{3}} .

4.2 Sum of divisors function

The sum of divisors functions is defined by σ(n) = ∑_{d|n} d.

Theorem 4.

∑_{n≤x} t_{Q}(n)σ(n) = \frac{π^{2}}{12}x^{2} (2 ∏_{q∈Q} \frac{(q − 1)^{2}(q + 1)}{q^{3}} − 1) + O (x(log x)^{2/3}) ,  (6)

and

∑_{n≤x} t_{Q}(n)\frac{1}{σ(n)} = E((\frac{2}{∏_{i} K_{q_{i}}} − 1)(log x + γ + F)

+ \frac{2}{∏_{i} K_{q_{i}}} · ∑_{i=1}^{r} \frac{log q_{i} · K′_{q_{i}}}{K_{q_{i}}} ) + O(x^{−1}(log x)^{2/3+r}(log log x)^{4/3}),

(7)

where γ is Euler’s constant, the constants K_{q} and K′_{q} are defined by

K_{q} = 1 + (q − 1) ∑_{v=1}^{∞} \frac{1}{q^{v+1} − 1} and K′_{q} = (q − 1) ∑_{v=1}^{∞} \frac{v}{q^{v+1} − 1},

(as a particular case, we have that K_{2} ≐ 1.606695 is the Erdős-Borwein constant, which can be seen in the sequence \underline{A065442} of the Sloane’s On-line Encyclopedia of Integer Sequences (OEIS) [8]), and the constants E and F are defined by

E = ∏_{p∈ℙ} α(p), F = ∑_{p∈ℙ} \frac{(p − 1)^{2}β(p) log p}{pα(p)},

with

α(p) = 1 − \frac{(p − 1)^{2}}{p} ∑_{j=1}^{∞} \frac{1}{(p^{j} − 1)(p^{j+1} − 1)},

β(p) = ∑_{j=1}^{∞} \frac{j}{(p^{j} − 1)(p^{j+1} − 1)}.

Proof. Concerning (6), we know from Walfisz [12, p. 99] that

∑_{n≤x} σ(n) = \frac{π^{2}}{12}x^{2} + O(x(log x)^{2/3}).

Then we have that R_{φ}(x) = (log x)^{2/3}, so condition (i) of Theorem 1 is satisfied.

On the other hand, we have that

S_{σ,q}(x) = 1 + ∑_{v=1}^{∞} σ(q^{v})x^{v} = 1 + ∑_{v=1}^{∞} \frac{q^{v+1} − 1}{q − 1}x^{v}

= 1 + \frac{1}{q − 1} (\frac{q^{2}x}{1 − qx} − \frac{x}{1 − x}) = \frac{1}{(1 − qx)(1 − x)}, |x| < 1/q,

so condition (ii) of Theorem 1 is satisfied.

Then we see that

\overline{S}_{σ,q}(x) = (1 − qx)(1 − x) = 1 − (q + 1)x + qx^{2} (x ∈ ℝ),

so b_{0,q} = 1, b_{1,q} = −(q + 1), b_{2,q} = q and b_{v,q} = 0 if v ≥ 3, and condition (iii) of Theorem 1 is satisfied. Furthermore, we see that S_{σ,q}(1/q^{2}) = \frac{1}{(1−1/q)(1−1/q^{2})} = \frac{q^{3}}{(q−1)^{2}(q+1)}.

Concerning (7), we know from Sita Ramaiah and Suryanarayana [6, Cor. 4.1] that

∑_{n≤x} \frac{1}{σ(n)} = E(log x + γ + F) + O(x^{−1}(log x)^{2/3}(log log x)^{4/3}).

Then we conclude that R_{1/φ}(x) = (log x)^{2/3}(log log x)^{4/3}, so condition (i) of Theorem 2 is satisfied.

On the other hand,

S_{1/σ,q}(x) = 1 + ∑_{v=1}^{∞} \frac{1}{σ(q^{v})}x^{v} = 1 + ∑_{v=1}^{∞} \frac{(q − 1)}{q^{v+1} − 1}x^{v}, (|x| < q),

so condition (ii) of Theorem 2 is satisfied.

The coefficients (\frac{q−1}{q^{v+1}−1}) of this last power series form a log-convex sequence. Indeed,

(\frac{q − 1}{q^{v+1} − 1})^{2} ≤ (\frac{q − 1}{q^{v} − 1}) · (\frac{q − 1}{q^{v+2} − 1}) , v ≥ 0

⇔ (q^{v} − 1)(q^{v+2} − 1) ≤ (q^{v+1} − 1)^{2} ⇔ 2q^{v+1} ≤ q^{v} + q^{v+2} ⇔ 2q ≤ 1 + q^{2}.

By Kaluza’s theorem (if a power series ∑_{v=0}^{∞} a_{v} satisfies the conditions a_{v} > 0 and a_{v}^{2} ≤ a_{v−1}a_{v+1}(v ≥ 1), then the coefficients of its reciprocal power series ∑_{v≥0} b_{v} satisfy −a_{v}/a_{0}^{2} ≤ b_{v} ≤ 0, v ≥ 1) [11, Lem. 8], we have that −\frac{q−1}{q^{v+1}−1} ≤ b_{v,q} ≤ 0, and therefore,

b_{v,q} ≪ (\frac{1}{q})^{v}. We conclude that M_{i} = \frac{1}{q_{i}} < \frac{q_{min}}{q_{i}}, so condition (iii) of Theorem 2 is satisfied with max(q_{i}M_{i}) = 1.

Furthermore, we see that

S_{1/σ,q}(1) = K_{q} and S′_{1/σ,q}(1) = K′_{q}.

4.3 Unitary divisor function

A natural number m is a unitary divisor of a number n if m is a divisor of n, and m and n/m are coprime. Let us define the arithmetic function σ^{∗}(n) as the sum of the unitary divisors of n (analogous to the sum of divisors function).

We have that σ^{∗} is multiplicative and σ^{∗}(p^{v}) = p^{v} + 1.

Theorem 5. If q_{min} ≥ q_{max}^{2/3} (in particular, if Q consists of a single prime), then

∑_{n≤x} t_{Q}(n)σ^{∗}(n) = \frac{π^{2}}{12ζ(3)}x^{2} (2 ∏_{q∈Q} \frac{(q^{2} − 1)}{(q^{2} + q + 1)} − 1) + O (x(log x)^{5/3}) ,  (8)

and

∑_{n≤x} t^{Q}(n)\frac{1}{σ^{∗}(n)} = E∗ Q log x + F ∗ Q + O(x^{−1}(log x)^{5/3+r}(log log x)^{4/3})  (9)

for some constant F_{Q}^{∗}, and for E_{Q}^{∗}, the latter being defined by

E_{Q}^{∗} = B^{∗} (\frac{2}{∏_{q∈Q} R_{q}} − 1) with B^{∗} = ∏_{p} (R_{p} · (1 − \frac{1}{p})) and R_{q} := 1+∑_{v=1}^{∞} \frac{1}{q^{v} + 1}.

Proof. We first prove (8). We know from Sitaramachandrarao and Suryanarayana [3, Eq. 1.4] that

∑_{n≤x} σ^{∗}(n) = \frac{π^{2}}{12ζ(3)}x^{2} + O(x(log x)^{5/3}),

so R_{σ^{∗}}(x) = (log x)^{5/3}, and condition (i) of Theorem 1 is satisfied.

Similarly, we have that

S_{σ^{∗},q}(x) = 1 + ∑_{v=1}^{∞} σ^{∗}(q^{v})x^{v} = 1 + ∑_{v=1}^{∞}(q^{v} + 1)x^{v} = \frac{1 − qx^{2}}{(1 − qx)(1 − x)}

for |x| < 1/q, so condition (ii) of Theorem 1 is satisfied.

We conclude that the reciprocal of the power series is given by

\overline{S}_{σ^{∗},q}(x) = \frac{1 − (q + 1)x + qx^{2}}{1 − qx^{2}} = −1 + \frac{2 − (q + 1)x}{1 − qx^{2}}

= −1 + (\frac{1 + 1/2(q^{1/2} + q^{−1/2})}{1 + \sqrt{q}x} + \frac{1 − 1/2(q^{1/2} + q^{−1/2})}{1 − \sqrt{q}x})

= −1 + (1 + 1/2(q^{1/2} + q^{−1/2})) ∑_{v=0}^{∞}(−1)^{v}\sqrt{q}^{v}x^{v} + (1 − 1/2(q^{1/2} + q^{−1/2})) ∑_{v=0}^{∞} \sqrt{q}^{v}x^{v},

with |x| < \frac{1}{\sqrt{q}}. Then

b_{v,q} = (1 + 1/2(q^{1/2} + q^{−1/2})) (−1)^{v}\sqrt{q}^{v} + (1 − 1/2(q^{1/2} + q^{−1/2})) \sqrt{q}^{v} ≪ \sqrt{q}^{v}.

We see that r_{i} = q_{i}^{0.5} and r_{i} ≤ q_{i}^{2}/q_{max}, so condition (iv) of Theorem 2 is satisfied. Furthermore, we see that

S_{σ^{∗},q}(\frac{1}{q^{2}} ) = \frac{1 − 1/q^{3}}{(1 − 1/q)(1 − 1/q^{2})} = \frac{q^{2} + q + 1}{q^{2} − 1}.

Now, we prove (9). We know from Sita Ramaiah and Suryanarayana [7, p. 1352] that

∑_{n≤x} \frac{1}{σ^{∗}(n)} = B^{∗} log x + D^{∗} + O(x^{−1}(log x)^{5/3}(log log x)^{4/3}).

Then we have R_{1/σ^{∗}} = (log x)^{5/3}(log log x)^{4/3}, so condition (i) of Theorem 2 is satisfied. On the other hand,

S_{1/σ^{∗},q}(x) = 1 + ∑_{v=1}^{∞} \frac{1}{σ^{∗}(q^{v})}x^{v} = 1 + ∑_{v=1}^{∞} \frac{1}{q^{v} + 1}x^{v} (|x| < q),

so condition (ii) of Theorem 2 is satisfied.

Let us choose a_{n} := \frac{1}{q^{n}+1}, if n ≥ 1, and a_{0} = 1, as the coefficients for this formal power series. We also define b_{n} as the coefficients of the associated reciprocal series. Then

a_{n}b_{n+1} = ∑_{k=1}^{n−1} b_{k}(a_{n+1}a_{n−k} − a_{n}a_{n+1−k}) + b_{n}(a_{n+1} − a_{n}a_{1}) (n ≥ 2),

\frac{1}{q^{n} + 1}b_{n+1} = ∑_{k=1}^{n−1} b_{k} · (\frac{1}{q^{n+1} + 1} · \frac{1}{q^{n−k} + 1} − \frac{1}{q^{n} + 1} · \frac{1}{q^{n+1−k} + 1})

+ b_{n} (\frac{1}{q^{n+1} + 1} − \frac{1}{q^{n} + 1} \frac{1}{q + 1})

= ∑_{k=1}^{n−1} b_{k} \frac{((q^{n} + 1)(q^{n+1−k} + 1) − (q^{n+1} + 1)(q^{n−k} + 1))}{(q^{n+1} + 1)(q^{n−k} + 1)(q^{n} + 1)(q^{n+1−k} + 1)} + b_{n} \frac{(q^{n} + 1)(q + 1) − (q^{n+1} + 1)}{(q^{n+1} + 1)(q^{n} + 1)(q + 1)}.

Then

\begin{aligned}b_{n+1} & = \frac{q^{n}(q − 1)}{q^{n+1} + 1} · ∑_{k=1}^{n−1} \frac{b_{k}(q^{−k} − 1)}{(q^{n−k} + 1)(q^{n+1−k} + 1)} + \frac{b_{n}}{q + 1} · \frac{(q^{n} + q)}{(q^{n+1} + 1)} \\ & = \frac{(q − 1)q^{n}}{q^{n+1} + 1} · ∑_{k=1}^{n−1} b_{k} \frac{q^{k} − q^{2k}}{(q^{n} + q^{k})(q^{n+1} + q^{k})} + \frac{b_{n}}{q + 1} · \frac{(q^{n} + q)}{(q^{n+1} + 1)}. \end{aligned}  (10)

Let us suppose that |b_{i}| ≤ \frac{C}{q^{i}} for some constant C and for all i = 0, 1, 2, …, n. We prove by induction that |b_{n+1}| ≤ \frac{C}{q^{n+1}} . Indeed, we have that

|b_{n+1}| ≤ C · (\frac{(q − 1)q^{n}}{q^{n+1} + 1} · ∑_{k=1}^{n−1} \frac{q^{k}}{(q^{k} + q^{n})^{2}} + \frac{q^{n} + q}{(q + 1) · q^{n} · (q^{n+1} + 1)}) .  (11)

Let us define the function f(n) := \frac{q^{n+1}·(q^{n}+q)}{(q+1)·q^{n}·(q^{n+1}+1)} = \frac{q^{n+1}+q^{2}}{(q+1)(q^{n+1}+1)} → \frac{1}{q+1} if n → ∞.

Furthermore, f(n) = \frac{1}{q+1} (1 + \frac{q^{2}−1}{q^{n+1}+1}) is an increasing function on n, therefore f(n) ≤ \frac{1.01}{q+1} for n large enough.

Similarly, we define the function g(n) := \frac{(q−1)q^{n}q^{n+1}}{q^{n+1}+1} · ∑_{k=1}^{n−1} \frac{q^{k}}{(q^{k}+q^{n})^{2}} . We have that

g(n) = \frac{(q − 1)q^{2n+1}}{q^{2n+1} + q^{n}} · ∑_{k=1}^{n−1} \frac{q^{k}}{(q^{n/2} + q^{k−n/2})^{2}} = \frac{q − 1}{1 + q^{−1−n}} · ∑_{k=1}^{n−1} \frac{1}{(q^{(n−k)/2} + q^{−(n−k)/2})^{2}}

= \frac{1}{4} \frac{(q − 1)}{(1 + q^{−1−n})} · ∑_{k=1}^{n−1} \frac{1}{cosh^{2}(\frac{n−k}{2} log q)} = \frac{(q − 1)}{4(1 + q^{−1−n})} · ∑_{j=1}^{n−1} \frac{1}{cosh^{2}(j log q/2)}

≤ (q − 1) ∑_{j=1}^{∞} \frac{1}{(e^{j log q/2} + e^{−j log q/2})^{2}} = (q − 1) · ∑_{j=1}^{∞} \frac{1}{(q^{j/2} + q^{−j/2})^{2}}

= (q − 1) ∑_{j=1}^{∞} \frac{q^{−j}}{(1 + q^{−j})^{2}} ≤ (q − 1) ∑_{j=1}^{∞} q^{−j}(1 − q^{−j} + q^{−2j})^{2}

= (x^{−1} − 1) ∑_{j=1}^{∞} x^{j} · (1 − x^{j} + x^{2j})^{2},

where we set x := q^{−1}. We have that

g(n) ≤ (q − 1) · ∑_{j=1}^{∞}(x^{j} − 2x^{2j} + 3x^{3j} − 2x^{4j} + x^{5j})

= (q − 1) (\frac{x}{1 − x} − \frac{2x^{2}}{1 − x^{2}} + \frac{3x^{3}}{1 − x^{3}} − \frac{2x^{4}}{1 − x^{4}} + \frac{x^{5}}{1 − x^{5}} )

= \frac{q^{9} + q^{8} + 5q^{7} + 5q^{6} + 7q^{5} + 7q^{4} + 8q^{3} + 4q^{2} + 3q + 1}{(q^{4} + q^{3} + q^{2} + q + 1)(q^{2} + q + 1)(q^{2} + 1)(q + 1)} =: s(q).

Continuing with (10), we have that, for all n sufficiently large,

|b_{n+1}| ≤ \frac{C}{q^{n+1}} (s(q) + \frac{1.01}{q + 1}) ≤ \frac{C}{q^{n+1}} ,

since s(q) + 1.01/(q + 1) < 1 for all primes q. Indeed, the real function s(x) + 1.01/(x + 1) decreases in the interval [2, δ], for some δ, and increases in [δ, ∞), but it is always less than 1. Then condition condition (iii) of Theorem 2 is satisfied with M_{i} = \frac{1}{q_{i}}.

The result (9) improves the result (51) by Tóth [11], thus solving open problem 41 of that publication.

4.4 Dedekind ψ function

Recall that the Dedekind function ψ(n) is defined as ψ(n) = n ∏_{p|n}(1 + \frac{1}{p}).

Theorem 6. We have that

∑_{n≤x} t_{Q}(n)ψ(n) = \frac{15}{2π^{2}} x^{2} (2 ∏_{q∈Q} (\frac{q(q − 1)}{q^{2} + 1}) − 1) + O(x(log x)^{2/3})  (12)

and

∑_{n≤x} t_{Q}(n)\frac{1}{ψ(n)} = C((log x + γ + D) · (2 ∏_{q∈Q} \frac{q^{2} − 1}{q^{2} + q − 1} − 1)

+ 2 ∏_{q∈Q} \frac{q^{2} − 1}{q^{2} + q − 1} · ∑_{q∈Q} \frac{q^{2} log q}{(q − 1)(q^{2} + q − 1)}) + O(x^{−1}(log x)^{2/3}(log log x)^{4/3}),

(13)

where

C = ∏_{p} (1 − \frac{1}{p(p + 1)}) and D = ∑_{p} \frac{log p}{p^{2} + p − 1}.

(The constant C ≐ 0.704442 is sometimes called the carefree constant, and its digits form the sequence \underline{A065463} in OEIS [8].)

Proof. The proof of (12) is quite similar to that of (4). We know from Walfisz [12, p. 100] that

∑_{n≤x} ψ(n) = \frac{15}{2π^{2}} x^{2} + O(x(log x)^{2/3}),

and we obtain that

S_{ψ,q}(x) := 1 + ∑_{v=1}^{∞} ψ(q^{v})x^{v} = \frac{1 + x}{1 − qx} (|x| < \frac{1}{q}),

thus concluding that

\overline{S}_{ψ,q}(x) = \frac{1 − qx}{1 + x} = 1 + (q + 1)x ∑_{v=1}^{∞}(−1)^{v}x^{v}.

Therefore, b_{v,q} = (−1)^{v}(q + 1) ≪ 1.

The proof of (13) is quite similar to that of (5). We know from Sita Ramaiah and Suryanarayana [7, Cor. 4.2] that

∑_{n≤x} \frac{1}{ψ(n)} = C(log x + γ + D) + O(x^{−1}(log x)^{2/3}(log log x)^{4/3}).

and, for the reciprocal power series, we obtain that

S_{1/ψ,q}(x) = 1 + ∑_{v=1}^{∞} \frac{1}{ψ(q^{v})}x^{v} = \frac{q^{2} + q − x}{(q + 1)(q − x)},

and

\overline{S}_{1/ψ,q}(x) = \frac{(q + 1)(q − x)}{q^{2} + q − x} = 1 − q ∑_{v=1}^{∞}(\frac{1}{q^{2} + q})^{v}x^{v}.

Then we have that b_{v,q} = −q (\frac{1}{q^{2}+q})^{v} and M_{i} = 1/(q_{i}^{2} + q_{i}).

4.5 Euler’s unitary function

We now consider an analogue of the Euler totient function, namely the multiplicative function φ^{∗} defined on the prime powers p^{v} by φ^{∗}(p^{v}) = p^{v} − 1.

Theorem 7. If q_{min} ≥ q_{max}^{2/3}, we have that

∑_{n≤x} t_{Q}(n)φ^{∗}(n) = \frac{C}{2} x^{2} (2 ∏_{q∈Q} (\frac{q^{2} − 1}{q^{2} + q − 1}) − 1) + O(x(log x)^{5/3}(log log x)^{4/3}), (14)

where C is defined as in Theorem 6.

Proof. The proof is quite similar to that of (8). We know from Sitaramachandrarao and Suryanarayana [3] that

∑_{n≤x} φ^{∗}(n) = \frac{C}{2} x^{2} + O(x(log x)^{5/3}(log log x)^{4/3}),

and we obtain that

S_{φ^{∗},q}(x) = 1 + ∑_{v=1}^{∞} φ^{∗}(q^{v})x^{v} = \frac{1 − 2x + qx^{2}}{(1 − qx)(1 − x)}

thus concluding that

\overline{S}_{φ^{∗},q}(x) = \frac{(1 − qx)(1 − x)}{qx^{2} − 2x + 1} = 1 + \frac{\sqrt{q − 1}}{2}i (−\frac{1}{ω} ∑_{v=0}^{∞}(x/ω)^{v+1} + \frac{1}{\overline{ω}} ∑_{v=0}^{∞}(x/\overline{ω})^{v+1}) ,

with ω = \frac{1}{q} · (1 + i\sqrt{q − 1}). Consequently, b_{v,q} = −\frac{\sqrt{q−1}}{2}i(−\frac{1}{ω}ω^{−v} − \frac{1}{\overline{ω}} \overline{ω}^{−v}) ≪ (q^{1/2})^{v}.

Note: Knowing, from [5], that for certain constants L^{∗} and M^{∗},

∑_{n≤x} \frac{1}{φ^{∗}(n)} = L^{∗} log x + M^{∗} + O(x^{−1}(log x)^{5/3}),

the author conjectures that a general result can be established for

∑_{n≤x} t_{Q}(n)\frac{1}{φ^{∗}(n)}

by proceeding in a manner similar to the one used in the proof of (9).

5 Acknowledgments

The author wishes to thank Josimar Ramirez for his thoughtful revision of this paper, and Diego Sejas for his suggestions on the writing of this work.

References

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[2] J. M. De Konick and A. Ivić, Topics in Arithmetical Functions, North-Holland Mathematics Studies 43, Notas de Matemática (72), North-Holland Publishing Company, 1980.

[3] R. Sitaramachandrarao and D. Suryanarayana, On ∑_{n≤x} σ^{∗}(n) and ∑_{n≤x} φ^{∗}(n), Proc. Amer. Math. Soc. 41 (1973), 61–66.

[4] R. Sitaramachandrarao, On error term of Landau, Indian J. Pure Appl. Math. 13 (1982), 882–885.

[5] V. Sita Ramaiah and M. V. Subbarao, Asymptotic formulae for sums of reciprocal of some multiplicative functions, J. Indian Math. Soc. 57 (1991), 153–167.

[6] V. Sita Ramaiah and D. Suryanarayana, Sums of reciprocal of some multiplicative functions, Math J. Okayama Univ. 21 (1979), 155–164.

[7] V. Sita Ramaiah and D. Suryanarayana, Sums of reciprocal of some multiplicative functions – II, Indian J. Pure Appl. Math. 11 (1980), 1334–1355.

[8] N. J. A. Sloane, The On-Line Encyclopedia of Integer Sequences, 2020. Available at

https://oeis.org.

[9] M. V. Subbarao, On some arithmetic convolutions, The Theory of Arithmetic Functions: Proceedings of the Conference at Western Michigan University, Lecture Notes in Mathematics, Vol. 251, Springer-Berlin, 1971, pp. 247–271.

[10] G. Tenenbaum, Introduction to Analytic and Probabilistic Number Theory, 3rd ed., Graduate Studies in Mathematics, Vol. 163, American Mathematical Society, 2015.

[11] L. Tóth, Alternating sums concerning multiplicative arithmetic functions, J. Integer Sequences 20 (2017), Article 17.2.1.

[12] A. Walfisz, Weylsche Exponentialsummen in der neueren Zahlentheorie, Mathematische Forschungsberichte, XV, VEB Deutscher Verlag der Wissenschaften, Berlin, 1963.

2010 Mathematics Subject Classification: Primary 11N37; Secondary 11A05, 11A25, 30B10. Keywords: multiplicative arithmetical function, generalized alternating sum, asymptotic formula, Euler’s totient function, sum of divisors function, divisor function.

(Concerned with sequences \underline{A065442} and \underline{A065463}.)

Received April 16 2020; revised versions received May 7 2020; August 24 2020. Published in Journal of Integer Sequences, October 17 2020.

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