A Note on the Average Order of the gcd-sum Function

Olivier Bordellès 2, Allée de la Combe La Boriette 43000 Aiguilhe France

                      borde43@wanadoo.fr

Abstract We prove an asymptotic formula for the average order of the gcd-sum function by

using a new convolution identity.

1 Introduction and main result

In 2001, Broughan [1] studied the gcd-sum function g defined for any positive integer n by

g (n) = ∑_{k=1}^{n} (k, n) ,

where (a, b) denotes the greatest common divisor of a and b. The author showed that g is multiplicative, and satisfies the convolution identity

g = φ ∗ Id,  (1)

where φ is the Euler totient function, Id is the completely multiplicative function defined by Id(n) = n and ∗ is the usual Dirichlet convolution product.

The function g appears in a specific lattice point problem [1, 6], where it can be used to estimate the number of integer coordinate points under the square-root curve. As a multiplicative function, the question of its average order naturally arises. By using the Dirichlet hyperbola principle, Broughan [1, Theorem 4.7] proved the following result: for any real number x ⩾ 1, the following estimate

∑_{n⩽x} g (n) = \frac{x^{2} log x}{2ζ (2)} + \frac{ζ (2)^{2}}{2ζ (3)}x^{2} + O (x^{3/2} log x)  (2)

holds.

The aim of this paper is to prove another convolution identity for g, and then get a fairly more precise estimate than (2) .

In what follows, τ is the well-known divisor function, μ is the Möbius function, 1 is the completely multiplicative function defined by 1 (n) = 1, F ∗ G is the Dirichlet convolution product of the arithmetical functions F and G, and we denote by θ the smallest positive real number such that

∑_{n⩽x} τ (n) = x log x + x (2γ − 1) + O_{ε} (x^{θ+ε})  (3)

holds for any real numbers x ⩾ 1 and ε > 0. The following inequality

θ ⩾ \frac{1}{4}

is well-known [3]. On the other hand, Huxley [4] showed that

θ ⩽ \frac{131}{416} ≈ 0.3149 …

holds. Now we are able to prove the following result

Theorem 1.1. For any real numbers x ⩾ 1 and ε > 0, we have

∑_{n⩽x} g (n) = \frac{x^{2} log x}{2ζ (2)} + \frac{x^{2}}{2ζ (2)} (γ − \frac{1}{2} + log (\frac{A^{12}}{2π} )) + O_{ε} (x^{1+θ+ε})

where A ≈ 1.282 427 129 … is the Glaisher-Kinkelin constant.

For further details about the Glaisher-Kinkelin constant, see [2, 5]. The reader interested in gcd-sum integer sequences should refer to Sloane’s sequence \underline{A018804}.

2 A convolution identity

The proof uses the following lemmas.

Lemma 2.1. For any real number z ⩾ 1 and any ε > 0, we have

∑_{n⩽z} nτ (n) = \frac{z^{2}}{2} log z + z^{2} (γ − \frac{1}{4}) + O_{ε} (z^{1+θ+ε}) .

Proof. The result follows easily from (3) and Abel’s summation. Lemma 2.2. We have

g = μ ∗ (Id · τ) .

Proof. Since φ = μ∗ Id, we have, using (1) ,

g = φ ∗ Id = μ ∗ (Id ∗ Id) = μ ∗ (Id · τ)

which is the desired result.

3 Proof of Theorem 1.1

By using Lemma 2.2, we get

∑_{n⩽x} g (n) = ∑_{d⩽x} μ (d) ∑_{k⩽x/d} kτ (k)

and Lemma 2.1 applied to the inner sum gives

∑_{n⩽x} g (n) = ∑_{d⩽x} μ (d) \{\frac{x^{2}}{d^{2}} (\frac{1}{2} log (\frac{x}{d}) + γ − \frac{1}{4}) + O_{ε} ((\frac{x}{d})^{1+θ+ε})\}
= x^{2} \{(\frac{1}{2} log x + γ − \frac{1}{4}) ∑_{d⩽x} \frac{μ (d)}{d^{2}} − ∑_{d⩽x} \frac{μ (d) log d}{2d^{2}}\} + O_{ε} (x^{1+θ+ε} ∑_{d⩽x} \frac{1}{d^{1+θ+ε}} )
= x^{2} \{(\frac{1}{2} log x + γ − \frac{1}{4}) ∑_{d=1}^{∞} \frac{μ (d)}{d^{2}} − ∑_{d=1}^{∞} \frac{μ (d) log d}{2d^{2}} + O (\frac{log x}{x})\} + O_{ε} (x^{1+θ+ε}) .

Now it is well-known that, for s ∈ ℂ such that Re s > 1, we have

\frac{1}{ζ (s)} = ∑_{d=1}^{∞} \frac{μ (d)}{d^{s}}

which gives by differentiation

\frac{ζ′ (s)}{(ζ (s))^{2}} = ∑_{d=1}^{∞} \frac{μ (d) log d}{d^{s}}

for Re s > 1, and hence

∑_{n⩽x} g (n) = \frac{x^{2}}{2ζ (2)} (log x − \frac{ζ′ (2)}{ζ (2)} + 2γ − \frac{1}{2}) + O_{ε} (x^{1+θ+ε}) ,

and we use

\frac{ζ′ (2)}{ζ (2)} = γ − log (\frac{A^{12}}{2π} ) .

The proof of the theorem is now complete.

Acknowledgment. I am grateful to the referee for his valuable suggestions on this paper.

References

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

[2] S. R. Finch, Mathematical Constants, Cambridge University Press, 2003, pp. 135–145.

[3] G. H. Hardy, The average order of the arithmetical functions P (x) and Δ (x), Proc. London Math. Soc. 15 (2) (1916), 192–213.

[4] M. N. Huxley, Exponential sums and lattice points III, Proc. London Math. Soc. 87 (2003), 591–609.

[5] H. Kinkelin, Über eine mit der Gammafunktion verwandte Transcendente und deren Anwendung auf die Integralrechnung, J. Reine Angew. Math. 57 (1860), 122–158.

[6] A. D. Loveless, The general GCD-product function, Integers 6 (2006), article A19, avail-

   able at http://www.integers-ejcnt.org/vol6.html. Corrigendum: 6 (2006), article

A39.

2000 Mathematics Subject Classification: Primary 11A25; Secondary 11N37. Keywords: gcd-sum function, Dirichlet convolution, average order of multiplicative functions .

(Concerned with sequence \underline{A018804}.)

Received July 10 2006; revised version received March 28 2007. Published in Journal of Integer Sequences, March 28 2007.

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