On the Lcm-Sum Function

Soichi Ikeda and Kaneaki Matsuoka Graduate School of Mathematics Nagoya University Furocho Chikusaku Nagoya 464-8602 Japan

                  m10004u@math.nagoya-u.ac.jp
                  m10041v@math.nagoya-u.ac.jp

Abstract We consider a generalization of the lcm-sum function, and we give two kinds of asymptotic formulas for the sum of that function. Our results include a generalization of Bordellès’s results and a refinement of the error estimate of Alladi’s result. We prove these results by the method similar to those of Bordellès.

1 Introduction

Pillai [6] first defined the gcd-sum function

g(n) = ∑_{j=1}^{n} gcd(j, n)

and studied this function. The function g(n) was defined again by Broughan [3]. Broughan considered

G_{α}(x) = ∑_{n≤x} n^{−α}g(n)

for α ∈ ℝ and x ≥ 1, and obtained some asymptotic formulas for G_{α}(x). The function G_{α}(x) was studied by some authors (see, for example, [4, 8]). Some generalizations of the function g(n) was considered (see, for example, [2, 10]).

On the other hand, the lcm-sum function

l(n) := ∑_{j=1}^{n} lcm(j, n)

was considered by some authors. Alladi [1] studied the sum

∑_{j=1}^{n}(lcm(j, n))^{r} (r ∈ ℝ, r ≥ 1)

and obtained

∑_{n≤x} ∑_{j=1}^{n}(lcm(j, n))^{r} = \frac{ζ(r + 2)}{2(r + 1)^{2}ζ(2)}x^{2r+2} + O(x^{2r+1+ϵ}).  (1)

We define the functions

L_{a}(n) := ∑_{j=1}^{n}(lcm(j, n))^{a}

T_{a}(x) := ∑_{n≤x} L_{a}(n)

for a ∈ ℤ and x ≥ 1.

Bordellès studied the sums T_{1}(x) and T_{−1}(x) and obtained

l(n) = \frac{1}{2}((Id^{2} · (φ + τ_{0})) ∗ Id)(n),

∑_{n≤x} ∑_{j=1}^{n} lcm(j, n) = \frac{ζ(3)}{8ζ(2)}x^{4} + O(x^{3}(log x)^{2/3}(log log x)^{4/3}) (x > e),

∑_{n≤x} ∑_{j=1}^{n} \frac{1}{lcm(j, n)} = \frac{(log x)^{3}}{6ζ(2)} + \frac{(log x)^{2}}{2ζ(2)} (γ + log(\frac{A^{12}}{2π} )) + O(log x),

where Id^{a}(n) = n^{a} (a ∈ ℤ),

τ_{0}(n) = \begin{cases}1, & if n = 1; \\ 0, & otherwise, \end{cases}

F ∗ G is the usual Dirichlet convolution product, and A is the Glaisher-Kinkelin constant [7, p. 25]). Gould and Shonhiwa [5] stated that the log-factors in the error term in the second formula can be removed.

In this paper we study T_{a}(x) for a ≥ 2 and a ≤ −2. The following theorems are our main results. These results are proved by the methods similar to those of Bordellès [2, Section 6].

We write f(x) = O(g(x)), or equivalently f(x) ≪ g(x), where there is a constant C > 0 such that |f(x)| ≤ Cg(x) for all values of x under consideration.

Theorem 1. Let B_{n} be Bernoulli numbers defined by

\frac{z}{e^{z} − 1} = ∑_{n=0}^{∞} B_{n} \frac{z^{n}}{n!} .

If we define

φ_{k}(n) := ∑_{d|n} μ(d)(\frac{d}{n})^{k}

and

M_{a}(n) := (Id^{2a} · (\frac{1}{a + 1}φ + \frac{1}{2}τ_{0} + \frac{1}{a + 1} ∑_{k=1}^{a−1} \binom{a + 1}{k + 1}B_{k+1}φ_{k}))(n),

then for a ∈ ℤ we have

L_{a}(n) = (M_{a} ∗ Id^{a})(n).

Theorem 2. Let x > e and a ∈ ℕ. Then we have

∑_{n≤x} L_{a}(n) = \frac{ζ(a + 2)}{2(a + 1)^{2}ζ(2)}x^{2a+2} + O(x^{2a+1}(log x)^{2/3}(log log x)^{4/3}) (as x → ∞),

where the implied constant depends on a.

Theorem 3. Let x ≥ 1 and k ∈ ℕ with k ≥ 2. Then we have

∑_{n=1}^{∞} L_{−k}(n) = \frac{ζ(k)}{2}(1 + \frac{ζ(k)^{2}}{ζ(2k)})  (2)

and

∑_{n≤x} L_{−k}(n) = \frac{ζ(k)}{2}(1 + \frac{ζ(k)^{2}}{ζ(2k)}) − \frac{ζ(k)x^{−k+1} log x}{(k − 1)ζ(k + 1)} + O(x^{−k+1}) (as x → ∞),

where the implied constant depends on k.

We note that the function L_{a}(n) is not multiplicative for all a ∈ ℤ \ {0}, but we can write L_{a}(n) (a ≥ 1) explicitly by Dirichlet convolution. In the proof of Theorem 2 we use this fact. The error estimates in Theorem 2 are better than (1). Since we have

g_{r}(n) := ∑_{j=1}^{n}(gcd(j, n))^{r} > φ(n)

for all r ∈ ℝ, the sum

∑_{n=1}^{∞} g_{r}(n)

is divergent for all r. Therefore the behavior of the sum T_{a}(x) (a ∈ ℤ and a ≤ −2) is completely different from that of the sum

∑_{n≤x} g_{a}(n).

2 Lemmas for the proof of theorems

In this section, we collect some auxiliary results and definitions.

Let B_{n}(x) be Bernoulli polynomials defined by

\frac{ze^{xz}}{e^{z} − 1} = ∑_{n=0}^{∞} B_{n}(x)\frac{z^{n}}{n!} .

The following relations are well-known [7, p. 59].

B_{n}(x + 1) − B_{n}(x) = nx^{n−1},

B_{n}(x) = ∑_{k=0}^{n} \binom{n}{k}B_{k}x^{n−k},

B_{n}(0) = B_{n}(1) = B_{n} (n > 1).

Lemma 4. Let m, n ∈ ℕ and

S_{n}(m) := ∑_{l=1}^{m} l^{n}.

Then we have

S_{n}(m) = \frac{m^{n+1}}{n + 1} + \frac{1}{2}m^{n} + \frac{1}{n + 1} ∑_{k=1}^{n−1} \binom{n + 1}{k + 1}B_{k+1}m^{n−k}.

Proof. We have

S_{n}(m) = \frac{1}{n + 1}(B_{n+1}(m + 1) − B_{n+1}(1))

= \frac{1}{n + 1}(B_{n+1}(m) + (n + 1)m^{n} − B_{n+1})

= \frac{1}{n + 1}(∑_{k=0}^{n+1} \binom{n + 1}{k}B_{k}m^{n+1−k} + (n + 1)m^{n} − B_{n+1})

= \frac{m^{n+1}}{n + 1} + \frac{1}{2}m^{n} + \frac{1}{n + 1} ∑_{k=1}^{n−1} \binom{n + 1}{k + 1}B_{k+1}m^{n−k}.

We use the following lemmas in the proof of Theorem 2. Lemma 5. Let r, k ∈ ℕ with r > k and x ≥ 1. We have

∑_{n≤x} n^{r}φ_{k}(n) ≤ x^{r+1}.

Proof. We have

∑_{n≤x} n^{r}φ_{k}(n) = ∑_{n≤x} n^{r−k} ∑_{d|n} μ(d)d^{k}

= ∑_{d≤x} μ(d)d^{k}(d^{r−k} + (2d)^{r−k} + ⋯ + (d⌊x/d⌋)^{r−k})

= ∑_{d≤x} μ(d)d^{r} ∑_{j≤x/d} j^{r−k}

≤ x^{r+1}.

Lemma 6. Let r ∈ ℕ and x > e. We have

∑_{n≤x} n^{r}φ(n) = \frac{x^{r+2}}{(r + 2)ζ(2)} + O(x^{r+1}(log x)^{2/3}(log log x)^{4/3}) (as x → ∞),

where the implied constant depends on r. Proof. We can obtain the lemma by the estimate [11]

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

and the partial summation formula.

3 Proof of Theorem 1 and Theorem 2

Proof of Theorem 1. We have

∑_{j=1}^{n}(\frac{j}{gcd(n, j)})^{a} = ∑_{d|n} \frac{1}{d^{a}} ∑_{\begin{matrix}j=1 \\ gcd(j,n)=d \end{matrix}}^{n} j^{a}

= ∑_{d|n} \frac{1}{d^{a}} ∑_{\begin{matrix}k≤n/d \\ gcd(k,n/d)=1 \end{matrix}}(kd)^{a} = ∑_{d|n} ∑_{\begin{matrix}k≤n/d \\ gcd(k,n/d)=1 \end{matrix}} k^{a}.

By Lemma 4 we have

∑_{\begin{matrix}k≤N \\ gcd(k,N)=1 \end{matrix}} k^{a} = ∑_{k≤N} k^{a} ∑_{d|gcd(k,N)} μ(d) = ∑_{d|N} d^{a}μ(d) ∑_{m≤N/d} m^{a}

= ∑_{d|N} d^{a}μ(d)(\frac{1}{a + 1}(\frac{N}{d} )^{a+1} + \frac{1}{2}(\frac{N}{d} )^{a}+

+ \frac{1}{a + 1} ∑_{k=1}^{a−1} \binom{a + 1}{k + 1}B_{k+1}(\frac{N}{d} )^{a−k})

= \frac{N^{a}}{a + 1} ∑_{d|N} μ(d)\frac{N}{d} + \frac{N^{a}}{2} ∑_{d|N} μ(d)+

+ \frac{N^{a}}{a + 1} ∑_{k=1}^{a−1} \binom{a + 1}{k + 1}B_{k+1} ∑_{d|N} μ(d)(\frac{d}{N} )^{k}

= N^{a}(\frac{1}{a + 1}φ + \frac{1}{2}τ_{0} + \frac{1}{a + 1} ∑_{k=1}^{a−1} \binom{a + 1}{k + 1}B_{k+1}φ_{k})(N).

Hence we obtain

L_{a}(n) = n^{a} ∑_{j=1}^{n}(\frac{j}{gcd(n, j)})^{a}

= ∑_{d|n}(\frac{n}{d})^{2a} · (\frac{1}{a + 1}φ + \frac{1}{2}τ_{0} + \frac{1}{a + 1} ∑_{k=1}^{a−1} \binom{a + 1}{k + 1}B_{k+1}φ_{k})(n/d) · d^{a}

= (M_{a} ∗ Id^{a})(n).

Proof of Theorem 2. By Lemma 5, Lemma 6 and Theorem 1, we have

∑_{n≤x} L_{a}(n) = ∑_{n≤x}(M_{a} ∗ Id^{a})(n) = ∑_{d≤x} d^{a} ∑_{m≤x/d} M_{a}(m)

= ∑_{d≤x} d^{a} ∑_{m≤x/d} m^{2a}(\frac{1}{a + 1}φ + \frac{1}{2}τ_{0} + \frac{1}{a + 1} ∑_{k=1}^{a−1} \binom{a + 1}{k + 1}B_{k+1}φ_{k})(m)

= \frac{1}{a + 1} ∑_{d≤x} d^{a} ∑_{m≤x/d} m^{2a}φ(m) + O(x^{a+1})+

+ \frac{1}{a + 1} ∑_{k=1}^{a−1} \binom{a + 1}{k + 1}B_{k+1} ∑_{d≤x} d^{a} ∑_{m≤x/d} m^{2a}φ_{k}(m)

= \frac{1}{a + 1} ∑_{d≤x} d^{a}(\frac{1}{(2a + 2)ζ(2)}(\frac{x}{d})^{2a+2}+

+ O((\frac{x}{d})^{2a+1}(log(x/d))^{2/3}(log log(x/d))^{4/3}))+

+ O(x^{a+1}) + O(∑_{d≤x} d^{a}(x/d)^{2a+1})

= \frac{x^{2a+2}}{(a + 1)(2a + 2)ζ(2)} ∑_{d≤x} \frac{1}{d^{a+2}} + O(x^{2a+1}(log x)^{2/3}(log log x)^{4/3})+

+ O(x^{2a+1}).

This implies the theorem.

4 Proof of Theorem 3

Proof of Theorem 3. Since we have

L_{−k}(n) = ∑_{j=1}^{n} \frac{1}{(lcm(n, j))^{k}} = \frac{1}{n^{k}} ∑_{j=1}^{n} \frac{(gcd(n, j))^{k}}{j^{k}} = \frac{1}{n^{k}} ∑_{d|n} d^{k} ∑_{\begin{matrix}j=1 \\ gcd(j,n)=d \end{matrix}}^{n} \frac{1}{j^{k}}

= \frac{1}{n^{k}} ∑_{d|n} d^{k} ∑_{\begin{matrix}i≤ \frac{n}{d} \\ gcd(i, \frac{n}{d})=1 \end{matrix}} \frac{1}{i^{k}d^{k}}

= \frac{1}{n^{k}} ∑_{d|n} ∑_{\begin{matrix}i≤ \frac{n}{d} \\ gcd(i, \frac{n}{d})=1 \end{matrix}} \frac{1}{i^{k}} ,

we obtain

∑_{n=1}^{∞} L_{−k}(n) = ∑_{n=1}^{∞} \frac{1}{n^{k}} ∑_{d|n} ∑_{\begin{matrix}i≤ \frac{n}{d} \\ gcd(i, \frac{n}{d})=1 \end{matrix}} \frac{1}{i^{k}} = ∑_{d=1}^{∞} ∑_{j=1}^{∞} \frac{1}{j^{k}d^{k}} ∑_{\begin{matrix}i≤j \\ gcd(i,j)=1 \end{matrix}} \frac{1}{i^{k}}

= ζ(k) ∑_{j=1}^{∞} \frac{1}{j^{k}} ∑_{\begin{matrix}i≤j \\ gcd(i,j)=1 \end{matrix}} \frac{1}{i^{k}}

= ζ(k) ∑_{n=1}^{∞} \frac{1}{n^{k}} ( ∑_{\begin{matrix}i≤j \\ gcd(i,j)=1 \\ ij=n \end{matrix}} 1).

Also we have

∑_{n=1}^{∞} \frac{1}{n^{k}} ( ∑_{\begin{matrix}i≤j \\ gcd(i,j)=1 \\ ij=n \end{matrix}} 1) = 1 + \frac{1}{2} ∑_{n=2}^{∞} \frac{1}{n^{k}} ( ∑_{\begin{matrix}gcd(i,j)=1 \\ ij=n \end{matrix}} 1)

and the relation [9, 1.2.8]

∑_{n=1}^{∞} \frac{1}{n^{k}} ( ∑_{\begin{matrix}gcd(i,j)=1 \\ ij=n \end{matrix}} 1) = \frac{ζ(k)^{2}}{ζ(2k)}.

Therefore we obtain

∑_{n=1}^{∞} \frac{1}{n^{k}} ( ∑_{\begin{matrix}i≤j \\ gcd(i,j)=1 \\ ij=n \end{matrix}} 1) = 1 + \frac{1}{2}(\frac{ζ(k)^{2}}{ζ(2k)} − 1) = \frac{1}{2}(1 + \frac{ζ(k)^{2}}{ζ(2k)}).

This implies (2).

By the relation

∑_{n≤x} L_{−k}(n) = \frac{ζ(k)}{2}(1 + \frac{ζ(k)^{2}}{ζ(2k)}) − ∑_{n>x} L_{−k}(n),

the remaining task is to estimate the sum ∑_{n>x} L_{−k}(n). We have

∑_{n>x} L_{−k}(n) = ∑_{n>x} \frac{1}{n^{k}} ∑_{d|n} ∑_{\begin{matrix}i≤ \frac{n}{d} \\ gcd(i, \frac{n}{d})=1 \end{matrix}} \frac{1}{i^{k}} = ∑_{d=1}^{∞} ∑_{h> \frac{x}{d}} \frac{1}{(hd)^{k}} ∑_{\begin{matrix}j≤h \\ gcd(j,h)=1 \end{matrix}} \frac{1}{j^{k}}

= ∑_{d=1}^{∞} ∑_{h> \frac{x}{d}} \frac{1}{(hd)^{k}} ∑_{j≤h} \frac{1}{j^{k}} ∑_{δ| gcd(j,h)} μ(δ)

= ∑_{d=1}^{∞} ∑_{h> \frac{x}{d}} \frac{1}{(hd)^{k}} ∑_{δ|h} ∑_{m≤ \frac{h}{δ}} \frac{μ(δ)}{m^{k}δ^{k}}

= ∑_{d=1}^{∞} \frac{1}{d^{k}} ∑_{δ=1}^{∞} ∑_{l> \frac{x}{dδ}} \frac{μ(δ)}{l^{k}δ^{2k}} ∑_{m≤l} \frac{1}{m^{k}}

= ∑_{q=1}^{∞} \frac{1}{q^{2k}} ∑_{dδ=q} d^{k}μ(δ) ∑_{l> \frac{x}{q}} \frac{1}{l^{k}} ∑_{m≤l} \frac{1}{m^{k}}

= ∑_{q=1}^{∞} \frac{1}{q^{2k}} ∑_{dδ=q} d^{k}μ(δ) ∑_{l> \frac{x}{q}} \frac{1}{l^{k}} (ζ(k) − \frac{l^{1−k}}{k − 1} + O(l^{−k}))

= ∑_{q<x} \frac{1}{q^{2k}} ∑_{dδ=q} d^{k}μ(δ) ∑_{l> \frac{x}{q}} \frac{1}{l^{k}} (ζ(k) − \frac{l^{1−k}}{k − 1} + O(l^{−k}))+

+ ∑_{q≥x} \frac{1}{q^{2k}} ∑_{dδ=q} d^{k}μ(δ) ∑_{l> \frac{x}{q}} \frac{1}{l^{k}} (ζ(k) − \frac{l^{1−k}}{k − 1} + O(l^{−k}))

=: S_{1} + S_{2},

say. We have

S_{1} = ∑_{q<x} \frac{1}{q^{2k}} ∑_{dδ=q} d^{k}μ(δ) ∑_{l> \frac{x}{q}} \frac{1}{l^{k}} (ζ(k) − \frac{l^{1−k}}{k − 1} + O(l^{−k}))

= ∑_{q<x} \frac{1}{q^{2k}} ∑_{dδ=q} d^{k}μ(δ)(\frac{ζ(k)}{k − 1}(x/q)^{−k+1} + O((x/q)^{−k})−

− \frac{(\frac{x}{q})^{−2k+2}}{(k − 1)(2k − 2)} + O((x/q)^{−2k+1}))

= ∑_{q<x} \frac{1}{q^{2k}} ∑_{dδ=q} d^{k}μ(δ)(\frac{ζ(k)}{k − 1}(x/q)^{−k+1} + O((x/q)^{−k}))

= ∑_{q<x} \frac{1}{q^{k}} ∑_{d|q} \frac{μ(d)}{d^{k}}(\frac{ζ(k)}{k − 1}(x/q)^{−k+1} + O((x/q)^{−k}))

= \frac{ζ(k)x^{−k+1}}{k − 1} ∑_{q<x} q^{−1} ∑_{d|q} \frac{μ(d)}{d^{k}} + O(x^{−k} ∑_{q<x} ∑_{d|q} \frac{|μ(d)|}{d^{k}})

= \frac{ζ(k)x^{−k+1}}{k − 1} ∑_{d<x} ∑_{j< \frac{x}{d}} j^{−1} \frac{μ(d)}{d^{k+1}} + O(x^{−k+1})

= \frac{ζ(k)x^{−k+1}}{k − 1} ∑_{d<x} log(\frac{x}{d})\frac{μ(d)}{d^{k+1}} + O(x^{−k+1})

= \frac{ζ(k)x^{−k+1} log x}{(k − 1)ζ(k + 1)} + O(x^{−k+1})

and

∑_{q≥x} \frac{1}{q^{2k}} ∑_{dδ=q} d^{k}μ(δ) ∑_{l> \frac{x}{q}} \frac{1}{l^{k}} (ζ(k) − \frac{l^{1−k}}{k − 1} + O(l^{−k}))

≪ ∑_{q≥x} \frac{1}{q^{2k}} ∑_{dδ=q} d^{k}μ(δ) ∑_{l=1}^{∞} \frac{1}{l^{k}} (ζ(k) − \frac{l^{1−k}}{k − 1} + O(l^{−k}))

≪ ∑_{q≥x} q^{−k} ∑_{d|q} \frac{|μ(d)|}{d^{k}}

≪ x^{−k+1}.

Therefore we obtain

∑_{n>x} L_{−k}(n) = \frac{ζ(k)x^{−k+1} log x}{(k − 1)ζ(k + 1)} + O(x^{−k+1}).

This completes the proof.

References

[1] K. Alladi, On generalized Euler functions and related totients, in New Concepts in Arithmetic Functions, Matscience Report 83, Madras, 1975.

[2] O. Bordellès, Mean values of generalized gcd-sum and lcm-sum functions, J. Integer Sequences 10 (2007), Article 07.9.2.

[3] K. A. Broughan, The gcd-sum function, J. Integer Sequences 4 (2001), Article 01.2.2.

[4] K. A. Broughan, The average order of the Dirichlet series of the gcd-sum function, J. Integer Sequences 10 (2007), Article 07.4.2.

[5] H. W. Gould and T. Shonhiwa, Functions of GCD’s and LCM’s, Indian J. Math. 39 (1997), 11–35.

[6] S. S. Pillai, On an arithmetic function, J. Annamalai Univ. 2 (1933), 243–248.

[7] H. M. Srivastava and J. Choi, Series Associated with Zeta and Related Functions, Kluwer Academic Publishers, 2001.

[8] Y. Tanigawa and W. Zhai, On the gcd-sum function, J. Integer Sequences 11 (2008), Article 08.2.3.

[9] E. C. Titchmarsh, The Theory of the Riemann Zeta-function, Second Edition, Oxford University Press, 1986.

[10] L. Tóth, A survey of gcd-sum functions, J. Integer Sequences 13 (2010), Article 10.8.1.

[11] A. Walfisz, Weylsche Exponentialsummen in der neueren Zahlentheorie, Leipzig BG Teubner, 1963.

2010 Mathematics Subject Classification: Primary 11A25; Secondary 11N37. Keywords: arithmetic function, lcm-sum function, least common multiple.

(Concerned with sequences \underline{A018804} and \underline{A051193}.)

Received August 19 2013; revised versions received November 14 2013; November 23 2013; December 19 2013. Published in Journal of Integer Sequences, December 27 2013.

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