The average order of the Dirichlet series of the gcd-sum function

Kevin A. Broughan

University of Waikato, Hamilton, New Zealand Version: 13th April 2007 E-mail: kab@waikato.ac.nz

The second term and improved error expressions for the partial sums of the Dirichlet series of the gcd-sum function, for all real values of the parameter, are derived using a result of Bordelles.

Key Words: greatest common divisor, Dirichlet series, gcd-sum

MSC2000 11A05, 11A25, 11M06, 11N37, 11N56.

Concerned with sequence A018804.

1. INTRODUCTION

The second term and improved error expressions for the partial sums of the Dirichlet series of the gcd-sum function, first given in [2], are derived, for all real values of the parameter, using a result of Bordelles [1].

Let (m, n) represent the positive greatest common divisor of the positive integers m and n, let g(n) := ∑_{i=1}^{n}(i, n) be the gcd-sum, let τ(n) be the number of divisors of n, let Id(n) := n be the identity function, for α ∈ ℝ let m_{α}(n) = 1/n^{α} be the monomial power function, and let ϕ(n) be Euler’s phi function. For x ≥ 1 let

G_{α}(x) := ∑_{n≤x} \frac{g(n)}{n^{α}}

be the average order.

Let θ be the smallest positive real number such that

∑_{n≤x} τ(n) = x log x + x(2γ − 1) + O_{ϵ}(x^{θ+ϵ}) (1)

for all ϵ > 0 and x ≥ 1. For a discussion about the value of θ and references see [1]. The current best value is slightly more than 1/4.

The Lemma of Bordelles [1] enables us to write g = μ ∗ (τ · Id) and this, combined with a best available error for the average order of τ(n), enables a further term and greater degree of precision to be obtained for the G_{α}(x) than the expression g = ϕ ∗ Id used in [2]. This is true for every value of α except for α = 2 in which case both expressions for g(n) give the same terms and error form.

In Section 3 we correct some errors in [2].

2. AVERAGE VALUES OF THE DIRICHLET SERIES

First we give two summations for the sum of divisor function τ(n).

Lemma 2.1. For all β satisfying β ≥ −θ and ϵ > 0:

∑_{n≤x} n^{β}τ(n) = \frac{x^{β+1}}{β + 1} log x + \frac{x^{β+1}}{β + 1}(2γ − \frac{1}{β + 1}) + O_{ϵ}(x^{θ+β+ϵ}).

Proof. This follows from (1) and Abel summation as in Lemma 1 of [1].

Lemma 2.2. For all β satisfying −1 < β < −θ:

∑_{n≤x} n^{β}τ(n) = \frac{x^{β+1}}{β + 1} log x + \frac{x^{β+1}}{β + 1}(2γ − \frac{1}{β + 1}) + O(1)

where the implied constant is absolute.

Let c_{1} := 2γ − \frac{1}{2} − \frac{ζ′(2)}{ζ(2)}. The theorem of Bordelles [1] can be written:

Lemma 2.3. For all ϵ > 0, as x → ∞

G_{0}(x) = \frac{x^{2} log x}{2ζ(2)} + c_{1} \frac{x^{2}}{2ζ(2)} + O_{ϵ}(x^{1+θ+ϵ}).

It is not needed in what follows, but for completeness the corresponding sum for the power of the multiplier -1 is [3]:

∑_{n≤x} \frac{τ(n)}{n} = \frac{log^{2} x}{2} + 2γ log x + γ^{2} − 2γ_{1} + O(\frac{1}{\sqrt{x}}),

where γ_{1} is the first Stieltjes constant.

Figure 1 is a plot of G_{0}(x) − \frac{x^{2} log x}{2ζ(2)} − c_{1} \frac{x^{2}}{2ζ(2)} for 1 ≤ x ≤ 1000. This should be compared with the corresponding plots for G_{α} given below. It

FIG. 1. Error value for G_{0}(x).

shows that, maybe, another continuous term can be extracted from the error. The existing computed error is large, but does, on the face of it, go to zero when divided by x^{2}, consistent with the given value of the constant c_{1}.

We can now improve each part of [2, Theorem 4.4]:

Theorem 2.1. For all ϵ > 0 as x → ∞

(1) If α ≤ 1 + θ:

G_{α}(x) = \frac{x^{2−α} log x}{(2 − α)ζ(2)} + \frac{x^{2−α}}{(2 − α)ζ(2)}(2γ − \frac{1}{2 − α} − \frac{ζ′(2)}{ζ(2)}) + O_{ϵ}(x^{θ+1−α+ϵ}).

(2) If 1 + θ < α < 2:

G_{α}(x) = \frac{x^{2−α} log x}{(2 − α)ζ(2)} + \frac{x^{2−α}}{(2 − α)ζ(2)}(2γ − \frac{1}{2 − α} − \frac{ζ′(2)}{ζ(2)}) + O(1).

(3) G_{2}(x) = \frac{log^{2} x}{2ζ(2)} + \frac{log x}{ζ(2)}(2γ − \frac{ζ′(2)}{ζ(2)}) + O(1).

(4) If α > 2:

G_{α}(x) = \frac{ζ(α − 1)^{2}}{ζ(α)} − \frac{log x}{(α − 2)ζ(2)x^{α−2}}

−\frac{1}{(α − 2)ζ(2)x^{α−2}} (2γ + \frac{1}{α − 2} − \frac{ζ′(2)}{ζ(2)}) + O_{ϵ}(\frac{1}{x^{α−1−θ−ϵ}}).

Proof.

(1) In this case α ≤ 1 + θ. Let β := 1 − α, c_{2} := 2γ − 1/(1 + β). Then, using Lemma 2.1 and the complete multiplicativity of m_{α}:

G_{α}(x) = ∑_{n≤x} \frac{g(n)}{n^{α}} = ∑_{n≤x}(m_{α} · g)(n)
= ∑_{n≤x}((m_{α} · μ) ∗ (m_{α} · Id · τ))(n)

= ∑_{d≤x} \frac{μ(d)}{d^{α}} ∑_{e≤x/d} e^{1−α}τ(e)

= ∑_{d≤x} \frac{μ(d)}{d^{α}}[\frac{1}{β + 1} \frac{x^{β+1}}{d^{β+1}} (log x − log d) + c_{2} \frac{x^{β+1}}{d^{β+1}} + O_{ϵ}((\frac{x}{d})^{θ+β+ϵ})]

= \frac{x^{β+1} log x}{β + 1} ∑_{d≤x} \frac{μ(d)}{d^{α+β+1}} − \frac{x^{β+1}}{β + 1} ∑_{d≤x} \frac{μ(d) log d}{d^{α+β+1}}

+c_{2}x^{β+1} ∑_{d≤x} \frac{μ(d)}{d^{α+β+1}} + O_{ϵ}(x^{θ+β+ϵ} ∑_{d≤x} \frac{1}{d^{θ+α+β+1}} )

Therefore

G_{α}(x) = \frac{x^{2−α} log x}{2 − α}[\frac{1}{ζ(2)} − ∑_{d>x} \frac{μ(d)}{d^{2}}] − \frac{x^{2−α}}{2 − α}[\frac{ζ′(2)}{ζ(2)^{2}} − ∑_{d>x} \frac{μ(d) log d}{d^{2}}]

+c_{2}x^{2−α}[\frac{1}{ζ(2)} − ∑_{d>x} \frac{μ(d)}{d^{2}}] + O_{ϵ}(x^{θ+ϵ+1−α})

= \frac{x^{2−α} log x}{(2 − α)ζ(2)} + \frac{x^{2−α}}{(2 − α)ζ(2)}[2γ − \frac{1}{2 − α} − \frac{ζ′(2)}{ζ(2)}]

+O_{ϵ}(x^{θ+ϵ+1−α})

(2) The derivation is the same as in (1), except we use Lemma 2.2 instead of Lemma 2.1.

(3) This is given in the proof of [2, Theorem 4.4, Case 2].

(4) Let α > 2. Then, using [2, Theorem 4.1] and Abel summation:

G_{α}(x) = \frac{ζ(α − 1)^{2}}{ζ(α)} − ∑_{n>x} \frac{g(n)}{n^{α}}

= \frac{ζ(α − 1)^{2}}{ζ(α)} − lim_{y→∞} \frac{G_{0}(y)}{y^{α}} + \frac{G_{0}(x)}{x^{α}} − α ∫_{x}^{∞} \frac{G_{0}(t)}{t^{α+1}}dt
= \frac{ζ(α − 1)^{2}}{ζ(α)} + \frac{G_{0}(x)}{x^{α}} − α ∫_{x}^{∞} \frac{G_{0}(t)}{t^{α+1}}dt
= \frac{ζ(α − 1)^{2}}{ζ(α)} − \frac{log x}{(α − 2)ζ(2)x^{α−2}}

−\frac{1}{2(α − 2)ζ(2)x^{α−2}} [2c_{1} + \frac{α}{α − 2}] + O_{ϵ}(\frac{1}{x^{α−1−θ−ϵ}}),

and the result follows after substituting for c_{1}.

FIG. 2. Error value for G_{α}(x) with α = 1.

Figure 2 is a plot of G_{1}(x)− \frac{x log x}{ζ(2)} − \frac{x}{ζ(2)}(2γ −1− \frac{ζ′(2)}{ζ(2)}) for 1 ≤ x ≤ 1000.

FIG. 3. Error value for G_{α}(x) with α = 3/2.

Figure 3 is a plot of

G_{α}(x) − \frac{x^{2−α} log x}{(2 − α)ζ(2)} − \frac{x^{2−α}}{(2 − α)ζ(2)}(2γ − \frac{1}{2 − α} − \frac{ζ′(2)}{ζ(2)})

for α = 3/2 and 1 ≤ x ≤ 1000.

3. ERRORS IN “THE GCD-SUM FUNCTION”

Corrections to some errors in [2]. Page 7 line 8: the second g should be Id, page 9 line 3: 5.1 should be 4.1, page 12 line 2: 5.4 should be 4.3, page 12 line 3: 5.3 should be 4.2, page 13 line 13 and line -1: e, d should be e.d, page 14 line -1: 5.4 should be 4.4.

acknowledgments: The support of the University of Waikato is gratefully acknowledged. The plots were produced using the Mathematica functions ListPlot, and DivisorSigma. Helpful suggestions made by a referee are also gratefully acknowledged.

REFERENCES

1. Bordellés, O. A note on the average order of the gcd-sum function, Journal of Integer Sequences, 10 (2007) Article 07.3.3.

2. Broughan, K.A. The gcd-sum function, Journal of Integer Sequences, 4 (2001), Article 01.2.2.

3. Riesel, H. and Vaughan, R.C. On sums of primes, Arkiv för Mathematik 21 (1983), 45-74.