arXiv:1711.04257v1 [math.NT] 12 Nov 2017
On multivariable averages of divisor functions
∗
L´aszl´o T´oth and Wenguang Zhai
Abstract P P We deduce asymptotic formulas for the sums n1 ,...,nr ≤x f (n1 · · · nr ) and n1 ,...,nr ≤x f ([n1 · · · nr ]), where r ≥ 2 is a fixed integer, [n1 , . . . , nr ] stands for the least common multiple of the integers n1 , . . . , nr and f is one of the divisor functions τ1,k (n) (k ≥ 1), τ (e) (n) and τ ∗ (n). Our formulas refine and generalize a result of Lelechenko (2014). A new generalization of the Busche-Ramanujan identity is also pointed out.
2010 Mathematics Subject Classification: 11A25, 11N37 Key Words and Phrases: divisor function, least common multiple, multiple Dirichlet series, average order, error term, Busche-Ramanujan identity
1
Introduction
P Let τ1,k (n) = abk =n 1, where k ∈ N := {1, 2, . . .} is a fixed integer. For k = 1 this is the divisor function τ (n). It is well known that X τ (n) = x log x + (2γ − 1)x + O(xθ+ε ), (1) n≤x
for any ε > 0, where γ is Euler’s constant and 1/4 ≤ θ < 1/3. The best result up to date, namely θ ≤ 131/416 = 0.3149 . . . is due to Huxley [3]. Furthermore, for k ≥ 2, X τ1,k (n) = ζ(k)x + ζ(1/k)x1/k + O(xθk +ε ), (2) n≤x
where 1/(2(k + 1)) ≤ θk ≤ 1/(k + 2), which can be improved. See, e.g., the book by Kr¨ atzel [4, 1057 = 0.2209 . . ., which is a result of Graham and Kolesnik [1]. Ch. 5]. Note that θ2 ≤ 4785 The exponential divisor function τ (e) is multiplicative and defined by τ (e) (pν ) = τ (ν) for every prime power pν (ν ∈ N). It is known that X τ (e) (n) = c1 x + c2 x1/2 + O(xλ+ε ), n≤x
for every ε > 0, where c1 , c2 are computable constants and λ = 1057 4785 = 0.2209 . . ., the error term being strongly related to the divisor function τ1,2 . See Wu [10]. ∗ This work is supported by the National Key Basic Research Program of China (Grant No. 2013CB834201). Part of this paper was completed while the first author visited the China University of Mining and Technology in November 2016.
1
Lelechenko [5] proved, using a multidimensional Perron formula and the complex integration method that X τ1,2 (mn) = A2 x2 + B2 x3/2 + O(x10/7+ε ), (3) m,n≤x
where A2 , B2 are constants and 10/7 = 1.4285 . . .. He noted that in the case k ≥ 3 the same method does not furnish the expected asymptotic formula X τ1,k (mn) = Ak x2 + Bk x1+1/k + O(xαk +ε ), (4) m,n≤x
since the obtained error term is larger than x4/3 , even under the Riemann hypothesis, and absorbs the term x1+1/k . It is also noted in paper [5] that formula (3) remains valid for the function τ (e) instead of τ1,2 , due to the fact that τ (e) (pν ) = τ1,2 (pν ) for ν ∈ {1, 2, 3, 4}. Another similar divisor function is τ ∗ (n) = 2ω(n) , representing the number of unitary divisors of n, which equals the number of squarefree divisors of n. One has X 2ζ ′ (2) 6 ∗ + O(R(x)), τ (n) = 2 x log x + 2γ − 1 − π ζ(2) n≤x
where R(x) ≪ x1/2 exp(−c0 (log x)3/5 (log log x)−1/5 ) with c0 a positive constant. See [6]. It is the goal of the present paper to improve the error term of (3) and to deduce formula P (4) with a sharp error term. P More generally, we derive asymptotic formulas for the sums τ (n · · · n ) and r n1 ,...,nr ≤x 1,k 1 n1 ,...,nr ≤x τ1,k ([n1 · · · nr ]), where k ≥ 1 and r ≥ 2 are fixed integers and [n1 , . . . nr ] stands for the least common multiple of n1 , . . . , nr . Furthermore, we deduce similar asymptotic formulas concerning the divisor functions τ (e) and τ ∗ . Our approach is based on the study of multiple Dirichlet series and the convolution method. A new generalization of the Busche-Ramanujan identity is also pointed out. P We remark that asymptotic formulas for sums n1 ,...,nr ≤x f ([n1 , . . . , nr ]), where f belongs P P to a large class of multiplicative functions, including σt (n) = d|n dt and φt (n) = d|n dt µ(n/d) with t ≥ 1/2 were established by Hilberdink and the first author [2]. However, those results can not be applied for the divisor functions investigated in the present paper.
2
Multiple Dirichlet series First we consider the function τ1,k .
Proposition 2.1. Let r ≥ 2, k ≥ 1 and let s1 , . . . , sr ∈ C with ℜsj > 1 (1 ≤ j ≤ r). Then ∞ X
n1 ,...,nr =1
τ1,k (n1 · · · nr ) = ζ(s1 )ζ(ks1 ) · · · ζ(sr )ζ(ksr )Fr,k (s1 , . . . , sr ), ns11 · · · nsrr
where Fr,k (s1 , . . . , sr ) =
∞ X
n1 ,...,nr
fr,k (n1 , . . . , nr ) ns11 · · · nsrr =1
is absolutely convergent provided that ℜsj > 0 (1 ≤ j ≤ r) and ℜ(sj + sℓ ) > 1 (1 ≤ j < ℓ ≤ r). 2
We remark that the following exact identity, valid for k = 1 was proved by the first author [7, Eq. (4.2)]: ∞ X
n1 ,...,nr
= ζ 2 (s1 ) · · · ζ 2 (sr )
Y p
τ (n1 · · · nr ) ns11 · · · nsrr =1
r X 1 + (−1)j−1 (j − 1) j=2
1
X
p 1≤i1 1, Then ∞ X
n1 ,n2 =1
τ1,k (n1 n2 ) = ζ(s1 )ζ(ks1 )ζ(s2 )ζ(ks2 )Fk (s1 , s2 ) ns11 ns22
where Fk (s1 , s2 ) =
Y p
1 +
k−1 X j=1
1 pjs1+(k−j)s2
−
k X j=1
1 pjs1+(k+1−j)s2
is absolutely convergent if and only if ℜ(js1 + (k − j)s2 ) > 1 (1 ≤ j ≤ k − 1) and ℜ(js1 + (k + 1 − j)s2 ) > 1 (1 ≤ j ≤ k). Corollary 1. Let k ≥ 1. For every n1 , n2 ∈ N, X fk (d1 , d2 )τ1,k (n1 /d1 )τ1,k (n2 /d2 ), τ1,k (n1 n2 ) = d1 |n1 d2 |n2
where the function fk (n1 , n2 ) is multiplicative and for prime powers pν1 , pν2 , if ν1 = ν2 = 0 or ν1 , ν2 ≥ 1 and ν1 + ν2 = k, 1, ν1 ν2 fk (p , p ) = −1, if ν1 , ν2 ≥ 1 and ν1 + ν2 = k + 1, 0, otherwise. Note that (5) is a generalization of the Busche-Ramanujan formula X τ (n1 n2 ) = µ(d)τ (n1 /d)τ (n2 /d), d|gcd(n1 ,n2 )
valid for every n1 , n2 ∈ N, which is recovered in the case k = 1. Proposition 2.3. Let r ≥ 2, k ≥ 1 and let s1 , . . . , sr ∈ C with ℜsj > 1 (1 ≤ j ≤ r). Then ∞ X
n1 ,...,nr =1
τ1,k ([n1 , . . . , nr ]) = ζ(s1 )ζ(ks1 ) · · · ζ(sr )ζ(ksr )Gr,k (s1 , . . . , sr ), ns11 · · · nsrr 3
(5)
where Gr,k (s1 , . . . , sr ) =
∞ X
n1 ,...,nr
gr,k (n1 , . . . , nr ) ns11 · · · nsrr =1
is absolutely convergent provided that ℜsj > 0 (1 ≤ j ≤ r) and ℜ(sj + sℓ ) > 1 (1 ≤ j < ℓ ≤ r). Concerning the exponential divisor function τ (e) we have Proposition 2.4. Let r ≥ 2 and let s1 , . . . , sr ∈ C with ℜsj > 1 (1 ≤ j ≤ r). Then ∞ X
n1 ,...,nr =1
and
∞ X
n1 ,...,nr =1
τ (e) (n1 · · · nr ) = ζ(s1 )ζ(2s1 ) · · · ζ(sr )ζ(2sr )Tr (s1 , . . . , sr ), ns11 · · · nsrr
τ (e) ([n1 , . . . nr ]) = ζ(s1 )ζ(2s1 ) · · · ζ(sr )ζ(2sr )Vr (s1 , . . . , sr ), ns11 · · · nsrr
where the Dirichlet series Tr (s1 , . . . , sr ) =
∞ X
t(n1 , . . . , nr ) ns11 · · · nsrr =1
∞ X
v(n1 , . . . , nr ) ns11 · · · nsrr
n1 ,...,nr
and Vr (s1 , . . . , sr ) =
n1 ,...,nr =1
are both absolutely convergent if and only if ℜsj > 1/5 (1 ≤ j ≤ r) and ℜ(sj + sℓ ) > 1 (1 ≤ j < ℓ ≤ r). Now we move to the function τ ∗ . Note that τ ∗ (n1 · · · nr ) = τ ∗ ([n1 , . . . , nr ]) for every n1 , . . . , nr ∈ N. We have the next result: Proposition 2.5. Let r ≥ 2 and let s1 , . . . , sr ∈ C with ℜsj > 1 (1 ≤ j ≤ r). Then ∞ X
n1 ,...,nr =1
τ ∗ (n1 · · · nr ) = ζ 2 (s1 ) · · · ζ 2 (sr )Hr (s1 , . . . , sr ), ns11 · · · nsrr
where Hr (s1 , . . . , sr ) =
Y p
1 1 − s1 p
1 1 1 · · · 1 − sr 2 − 1 − s1 · · · 1 − sr p p p
is absolutely convergent if and only if ℜsj > 1/2 (1 ≤ j ≤ r).
4
3
Asymptotic formulas We prove the following results.
Theorem 3.1. If k, r ≥ 2, then X τ1,k (n1 · · · nr ) = Ar,k xr + Br,k xr−1+1/k + O(xr−1+θk +ε ),
(6)
n1 ,...,nr ≤x
X
τ1,k ([n1 , . . . , nr ]) = Cr,k xr + Dr,k xr−1+1/k + O(xr−1+θk +ε ),
(7)
n1 ,...,nr ≤x
for every ε > 0, where θk is the exponent in the error term of formula (2) and Ar,k , Br,k , Cr,k , Dr,k are computable constants. Here Ar,k :=
Y
1−
p
Cr,k :=
Y p
1 p
1 1− p
r
r
∞ X
ν1 ,...,νr ∞ X
ν1 ,...,νr
⌊(ν1 + · · · + νr )/k⌋ + 1 , ν1 +···+νr p =0
⌊max(ν1 , . . . , νr )/k⌋ + 1 . pν1 +···+νr =0
It follows from Proposition 2.2 that for r = 2 and k ≥ 2, Y k−1 k 2 2 A2,k = ζ (k)Fk (1, 1) = ζ (k) 1 + k − k+1 . p p p Theorem 3.2. If r ≥ 2, then X τ (e) (n1 · · · nr ) = Kr xr + Lr xr−1/2 + O(xr−1+θ2 +ε ), n1 ,...,nr ≤x
X
τ (e) ([n1 , . . . , nr ]) = Kr′ xr + L′r xr−1/2 + O(xr−1+θ2 +ε ),
n1 ,...,nr ≤x
for every ε > 0, where θ2 ≤ 0.2209 . . . is defined by (2) and Kr , Lr , Kr′ , L′r are computable constants. Here ∞ Y X 1 r τ (ν1 + · · · + νr ) , 1 + 1− Kr = ν1 +···+νr p p p ν ,...,ν =0 1
r
ν1 +···+νr ≥1
Kr′ =
Y p
1−
1 p
r
1 +
∞ X
ν1 ,...,νr =0 ν1 +···+νr ≥1
τ (max(ν1 , . . . , νr )) . pν1 +···+νr
Our multivariable asymptotic formulas regarding the divisor functions τ and τ ∗ are special cases of the following general convolution result.
5
Theorem 3.3. Let r ≥ 2 and let h : Nr → C, g : Nr → C, fj : N → C (1 ≤ j ≤ r) be arithmetic functions such that X h(n1 , . . . , nr ) = g(d1 , . . . , dr )f1 (m1 ) · · · fr (mr ) d1 m1 =n1 ,...,dr mr =nr
for every n1 , . . . , nr ∈ N. Assume that there exist constants 0 < bj < aj (1 ≤ j ≤ r) such that X Fj (x) := fj (n) = xaj Pj (log x) + O(xbj ) (1 ≤ j ≤ r), n≤x
where Pj (u) are polynomials in u of degrees δj , with leading coefficients Kj (1 ≤ j ≤ r). Define ∆2 := max (aj − bj ), ∆1 := min (aj − bj ). 1≤j≤r
1≤j≤r
Suppose further that there exists a positive number δ > ∆2 such that the Dirichlet series G(s1 , . . . , sr ) :=
∞ X
n1 ,...,nr =1
g(n1 , . . . , nr ) ns11 · · · nsrr
is absolutely convergent when ℜ(s1 + · · · + sr ) > a1 + · · · + ar − δ. Then the asymptotic formula X h(n1 , . . . , nr ) = xa1 +···+ar Q(log x) + O(xa1 +···+ar −∆1 (log x)δ1 +···+δr ) n1 ,...,nr ≤x
holds, where Q(u) is a polynomial in u of degree δ1 + · · · + δr , with leading coefficient K1 · · · Kr G(a1 , . . . , ar ). Theorem 3.4. If r ≥ 2, then X
τ (n1 · · · nr ) = xr Pr (log x) + O(xr−1+θ+ε ),
n1 ,...,nr ≤x
X
τ ([n1 , . . . , nr ]) = xr Qr (log x) + O(xr−1+θ+ε ),
n1 ,...,nr ≤x
for every ε > 0, where θ is the exponent in (1), Pr (t) and Qr (t) are polynomials in t of degree r having the leading coefficients KP,r :=
Y p
and KQ,r :=
Y p
1 1− p
1 1− p
2r
r−1
∞ X
ν1 ,...,νr
respectively.
6
r−1 1+ p
max(ν1 , . . . , νr ) + 1 , pν1 +···+νr =0
(8)
Note that the constant KP,r defined by (8) equals the asymptotic density of the set of r-tuples of positive integers with pairwise relatively prime components. See [8, 9]. Furthermore, in the case r = 2, KQ,2 = ζ(2)KP,3 = ζ(2)
Y p
1 1− p
2
2 1+ p
.
Theorem 3.5. If r ≥ 2, then X τ ∗ (n1 · · · nr ) = xr Pr∗ (log x) + O(xr−1/2 (log x)r ), n1 ,...,nr ≤x
where Pr∗ (t) is a polynomials in t of degree r having the leading coefficient Y 1 r 1 r 2− 1− . KP ∗ ,r := 1− p p p It is easier to derive asymptotic formulas for similar sums involving the greatest common divisor (n1 , . . . , nr ). Namely, the identity X X f ((n1 , . . . , nr )) = (µ ∗ f )(d)⌊x/d⌋r n1 ,...,nr ≤x
d|n
leads to asymptotic formulas for various multiplicative arithmetic functions f in the case r ≥ 2. For example, if f = τ1,k , then we deduce the formula X τ1,k ((n1 , . . . , nr )) = ζ(kr)xr + O(Rr (x)), n1 ,...,nr ≤x
where Rr (x) = xr−1 (r ≥ 3), R2 (x) = x log x, valid for every k ≥ 1.
4
Proofs
Proof of Proposition 2.1. The function n 7→ τ1,k (n) is multiplicative and τ1,k (pν ) = ⌊ν/k⌋ + 1 for every prime power pν (ν ≥ 0). The function (n1 , . . . , nr ) 7→ τ1,k (n1 · · · nr ) is multiplicative, viewed as a function of r variables. Therefore, its multiple Dirichlet series can be expanded into an Euler product. See the survey paper [8] on general accounts concerning multiplicative functions of several variables. We obtain ∞ X
n1 ,...,nr
=
Y
∞ X
p ν1 ,...,νr
∞ τ1,k (n1 · · · nr ) Y X τ1,k (pν1 +···+νr ) = ns11 · · · nsrr pν1 s1 +···+νr sr p ν ,...,ν =0 =1 1
r
⌊(ν1 + · · · + νr )/k⌋ + 1 = ζ(s1 )ζ(ks1 ) · · · ζ(sr )ζ(ksr )Fr,k (s1 , . . . , sr ), pν1 s1 +···+νr sr =0
where Fr,k (s1 , . . . , sr ) 7
=
Y p
1 1 − s1 p
1
1 · · · 1 − sr p
∞ X
⌊(ν1 + · · · + νr )/k⌋ + 1 pν1 s1 +···+νr sr ν1 ,...,νr =0 Y 1 1 1 1 1 1 = 1 − s1 − ks1 + (k+1)s · · · 1 − sr − ksr + (k+1)s r 1 p p p p p p p 1−
pks1
∞ X
×
ν1 ,...,νr =0
p
1−
pksr
⌊(ν1 + · · · + νr )/k⌋ + 1 pν1 s1 +···+νr sr
Y 1 + =
1
∞ X
cν1 ,...,νr
ν1 ,...,νr =0 #A(ν1 ,...,νr )≥2
pν1 s1 +···+νr sr
with some coefficients cν1 ,...,νr , where A(ν1 , . . . , νr ) := {j : 1 ≤ j ≤ r, νj 6= 0}. Here the coefficient c of 1/pℓsj (the case νt = 0 for all t 6= j and νj = ℓ) is zero for every 1 ≤ j ≤ r and ℓ ≥ 1. Indeed, ℓ−1 ℓ if 1 ≤ ℓ ≤ k − 1, ⌊ k ⌋ + 1 − ⌊ k ⌋ + 1 = 0, k k−1 c= ⌊ k ⌋ + 1 − ⌊ k ⌋ + 1 − 1 = 0, if ℓ = k, ℓ ℓ−1 ℓ−k ℓ−k−1 ⌊ k ⌋ + 1 − ⌊ k ⌋ + 1 − ⌊ k ⌋ + 1 + ⌊ k ⌋ + 1 = 0, if ℓ ≥ k + 1.
Hence a sufficient condition of absolute convergence of Fr,k (s1 , . . . , sr ) is that formulated in the statement of this Proposition. Proof of Proposition 2.2. In the case r = 2 we have Dk (s1 , s2 ) :=
∞ X
n1 ,n2
=
∞ YX p j=0
∞ X
∞
τ1,k (n1 n2 ) Y X ⌊(ν1 + ν2 )/k⌋ + 1 = ns11 ns22 pν1 s1 +ν2 s2 p ν ,ν =0 =1 1
j+1
ν1 ,ν2 =0 jk≤ν1 +ν2 ≤(j+1)k−1
pν1 s1 +ν2 s2
(j+1)k−1 ∞ ℓ X YX 1 1 X . (j + 1) = ℓs2 ν (s 1 p p 1 −s2 ) p j=0 ν =0 ℓ=jk 1
Let x = p−s1 , y = p−s2 . We deduce that Dk (s1 , s2 ) = Sk (x, y) =
(j+1)k−1 ∞ X X (j + 1) = yℓ
=
x 1− y
Q
p Sk (x, y),
where
(j+1)k−1 ∞ ℓ a X X X x (j + 1) yℓ y a=0 j=0
j=0
2
ℓ=jk
ℓ=jk
ℓ+1 ! x −1 x 1− 1− y y
−1 X (j+1)k−1 (j+1)k−1 ∞ X X x (j + 1) xℓ yℓ − y j=0
ℓ=jk
8
ℓ=jk
=
1−
=
x y
−1
∞ yk X
1 − 1−y
1−
x y
−1
(j + 1)y jk −
j=0
∞ xk X
x 1− · y 1−x
j=0
(j + 1)xjk
(1 − x)(1 − xk ) − xy −1 (1 − y)(1 − y k ) (1 − x)(1 − y)(1 − xk )(1 − y k )
1 = (1 − x)(1 − y)(1 − xk )(1 − y k )
1+
k−1 X
xt y k−t −
t=1
This gives the result.
k X
!
xt y k+1−t .
t=1
Proof of Proposition 2.3. Similar to the proof of Proposition 2.1. The function (n1 , . . . , nr ) 7→ τ1,k ([n1 , . . . , nr ]) is also multiplicative. Its multiple Dirichlet series can be written as ∞ X
n1 ,...,nr
=
Y
∞ X
p ν1 ,...,νr
∞ τ1,k ([n1 , . . . , nr ]) Y X τ1,k (pmax(ν1 ,...,νr ) ) = ns11 · · · nsrr pν1 s1 +···+νr sr p ν ,...,ν =0 =1 1
r
⌊max(ν1 , . . . , νr )/k⌋ + 1 = ζ(s1 )ζ(ks1 ) · · · ζ(sr )ζ(ksr )Gr,k (s1 , . . . , sr ), pν1 s1 +···+νr sr =0
where
=
Y p
Gr,k (s1 , . . . , sr ) 1 1 1 1 1 1 1 − s1 − ks1 + (k+1)s · · · 1 − sr − ksr + (k+1)s r 1 p p p p p p ×
∞ X
ν1 ,...,νr =0
=
Y 1 + p
⌊max(ν1 , . . . , νr )/k⌋ + 1 pν1 s1 +···+νr sr ∞ X
ν1 ,...,νr =0 #A(ν1 ,...,νr )≥2
c′ν1 ,...,νr
pν1 s1 +···+νr sr
since here the coefficient c′ of 1/pℓsj equals the coefficient c of 1/pℓsj in Fr,k (s1 , . . . , sr ), which vanishes, as explained in the proof of Proposition 2.1. Proof of Proposition 2.4. Similar to the proof of Proposition 2.1. Since τ (e) (pν ) = τ (ν) = τ1,2 (pν ) for ν ∈ {1, 2, 3, 4}, in Tr (s1 , . . . , sr ) and Vr (s1 , . . . , sr ) the coefficients of the terms 1/pℓsj will be zero for every 1 ≤ j ≤ r and ℓ ∈ {1, 2, 3, 4}. However, τ (e) (p5 ) = τ (5) = 2 6= 3 = τ1,2 (p5 ), therefore the coefficients of the terms 1/p5sj will not vanish (they are −1 for every 1 ≤ j ≤ r). Hence for absolute convergence it is necessary that ℜsj > 1/5 (1 ≤ j ≤ r). Together with ℜ(sj + sℓ ) > 1 (1 ≤ j < ℓ ≤ r), these are necessary and sufficient conditions for absolute convergence.
9
Proof of Proposition 2.5. The function (n1 , . . . , nr ) 7→ τ ∗ (n1 · · · nr ) is multiplicative. Its multiple Dirichlet series can be written as ∞ X
n1 ,...,nr
Y 1 + = p
∞ X
∞ τ ∗ (n1 · · · nr ) Y X τ ∗ (pν1 +···+νr ) ) = ns11 · · · nsrr pν1 s1 +···+νr sr p ν ,...,ν =0 =1
! −1 −1 Y 1 1 = · · · 1 − sr −1 2 1 − s1 pν1 s1 +···+νr sr p p p 2
ν1 ,...,νr =0 ν1 +···+νr ≥1
r
1
= ζ 2 (s1 ) · · · ζ 2 (sr )Hr (s1 , . . . , sr ),
where in Hr (s1 , . . . , sr ) the coefficients of 1/psj are zero (1 ≤ j ≤ r). Proof of Theorem 3.1. We prove formula (6). According to Proposition 2.1, X τ1,k (n1 · · · nr ) = fr,k (d1 , . . . , dr )τ1,k (m1 ) · · · τ1,k (mr )
(9)
d1 m1 =n1 ,...,dr mr =nr
for every n1 , . . . , nr ∈ N, where fr,k is a multiplicative function and symmetric in the variables. Therefore, X
Sr,k (x) :=
X
τ1,k (n1 · · · nr ) =
n1 ,...,nr ≤x
fr,k (d1 , . . . , dr )
r Y
X
τ1,k (mj ).
j=1 mj ≤x/dj
d1 ,...,dr ≤x
For k ≥ 2 we deduce by (2) that Sr,k (x) =
X
fr,k (d1 , . . . , dr )
r Y
x ζ(k) + ζ(1/k) dj
j=1
d1 ,...,dr ≤x
x dj
1/k
θk +ε ! x + O( ) . dj
Here the main term will be X
Mr,k (x) := (ζ(k))r xr
d1 ,...,dr ≤x
= (ζ(k))r xr
∞ X
fr,k (d1 , . . . , dr ) d1 · · · dr
fr,k (d1 , . . . , dr ) + Rr,k (x) = (ζ(k))r xr Fr,k (1, . . . , 1) + Rr,k (x), d1 · · · dr
d1 ,...,dr =1
where Fr,k (1, . . . , 1) is convergent and its value is by (9), Fr,k (1, . . . , 1) =
Y p
−r
= (ζ(k))
Y p
1 1− p
1 1− p
r
r
1 1− k p
∞ X
ν1 ,...,νr
10
r
∞ X
ν1 ,...,νr =0
τ1,k (pν1 +···+νr ) pν1 +···+νr
⌊(ν1 + · · · + νr )/k⌋ + 1 , pν1 +···+νr =0
(10)
while Rr,k (x) ≪ xr P′
X′ |fr,k (d1 , . . . , dr )| , d1 · · · dr
meaning that d1 , . . . , dr ≤ x does not hold. That is, there is at least one t such that dt > x. We can assume, without restricting the generality, that t = 1. We obtain that X′ |fr,k (d1 , . . . , dr )| X′ |fr,k (d1 , . . . , dr )| 1 = · 1−ε ε d1 · · · dr d1 d2 · · · dr d1 d >x d >x 1
1
≪
1 x1−ε
∞ X
d1 ,...,dr =1
|fr,k (d1 , . . . , dr )| 1 ≪ 1−ε , ε d1 d2 · · · dr x
since the latter series is Fr,k (ε, 1, . . . , 1), which converges by Proposition 2.1. This gives Rr,k (x) ≪ xr−1+ε and Mr,k (x) = Ak,r xr + O(xr−1+ε ). (11) By multiplying in (10) the terms ζ(k) dxj (1 ≤ j ≤ r − 1) and ζ(1/k)( dxr )1/k we have (ζ(k))r−1 ζ(1/k)xr−1+1/k
X
d1 ,...,dr ≤x ∞ X
= (ζ(k))r−1 ζ(1/k)xr−1+1/k
fr,k (d1 , . . . , dr ) 1/k
d1 · · · dr−1 dr
fr,k (d1 , . . . , dr ) 1/k
d1 ,...,dr =1
d1 · · · dr−1 dr
+ Tr,k (x)
= (ζ(k))r−1 ζ(1/k)xr−1+1/k Fr,k (1, . . . , 1, 1/k) + Tr,k (x), where Fr,k (1, . . . , 1, 1/k) is convergent and Tr,k (x) ≪ xr−1+1/k with
P′
X′ |fr,k (d1 , . . . , dr )| 1/k
d1 · · · dr−1 dr
as above. There are two cases. Case I: Assuming that dr > x we deduce X′ |fr,k (d1 , . . . , dr )|
dr >x
1/k d1 · · · dr−1 dr
≪
1 x1/k−ε
∞ X
d1 ,...,dr
=
X′ |fr,k (d1 , . . . , dr )| 1 · 1/k−ε ε d1 · · · dr−1 dr dr dr >x
|fr,k (d1 , . . . , dr )| 1 ≪ 1/k−ε . d1 · · · dr−1 dεr x =1
Case II: If dr ≤ x, then there is a t ∈ {1, . . . , r − 1} such that dt > x. We deduce by taking t = 1, X′ |fr,k (d1 , . . . , dr )| d1−1/k X′ |fr,k (d1 , . . . , dr )| r = · 1−ε εd · · · d 1/k d d d r−1 r 1 2 1 d1 >x d1 >x d1 d2 · · · dr−1 dr dr ≤x
dr ≤x
≪
x1−1/k x1−ε
∞ X
d1 ,...,dr
|fr,k (d1 , . . . , dr )| 1 ≪ 1/k−ε . dε1 d2 · · · dr−1 dr x =1 11
Hence Tr,k (x) ≪ xr−1+ε , the same error as in (11). All the terms obtained by multiplying in (10) ζ(k) dxj (j ∈ {1, . . . , r} \ {t}) and ζ(1/k)( dxt )1/k are of the same size and give together Br,k xr−1+1/k + O(xr−1+ε ),
(12)
where Br,k = r(ζ(k))r−1 ζ(1/k)Fr,k (1, . . . , 1, 1/k). Now, if in (10) we take an error term, say O(( dxr )θk +ε ), then we have to consider ζ(k) dxj (1 ≤ j ≤ r − 1) to obtain, by multiplying, the largest term, which is X |fr,k (d1 , . . . , dr )| ≪ xr−1+θk +ε θk +ε d1 ,...,dr ≤x d1 · · · dr−1 dr ≪x
∞ X
r−1+θk +ε
|fr,k (d1 , . . . , dr )|
d1 ,...,dr =1
giving the error
d1 · · · dr−1 dθrk +ε
,
xr−1+θk +ε ,
(13)
since the involved series is convergent. Therefore, (6) follows by (11), (12) and (13). The proof of (7) is by similar arguments, based on Proposition 2.3. Proof of Theorem 3.2. Similar to the proof of Theorem 3.1 by selecting k = 2 and using the fact that the behavior of τ (e) is similar to τ1,2 (n), as explained before. Proof of Theorem 3.3. For each 1 ≤ j ≤ r, we write Fj (x) = Mj (x) + Ej (x), where Mj (x) = xaj Pj (log x), Ej (x) = O(xbj ). Then we have X
h(n1 · · · nr )
=
n1 ,...,nr ≤x
X
g(d1 , . . . , dr )
X
g(d1 , . . . , dr )
d1 ,...,dr ≤x
=
r Y
j=1 r Y
Fj (x/dj )
(14)
(Mj (x/dj ) + Ej (x/dj )) .
j=1
d1 ,...,dr ≤x
It is easy to see that we can write r Y
(Mj (x/dj ) + Ej (x/dj )) =
Mj (x/dj ) + η(x; d1 , . . . , dr ),
j=1
j=1
η(x; d1 , . . . , dr ) ≪
r Y
ak r X x bj Y x × (log x)δ1 +···+δr . dj dk j=1
1≤k≤r k6=j
12
(15)
Let Lj (x) := xa1 +···+aj−1 +bj +aj+1 +···+ar
(1 ≤ j ≤ r).
The contribution of η(x; d1 , . . . , dr ) is X ≪ |g(d1 , . . . , dr )| × |η(x; d1 , · · · , dr )|
(16)
d1 ,...,dr ≤x
δ1 +···+δr
≪ (log x)
r X
|g(d1 , . . . , dr )|
X
Lj (x)
a1 d1 ,...,dr ≤x d1 δ1 +···+δr
j=1
≪ xa1 +···+ar −∆1 (log x)
a
b
a
j−1 j+1 · · · dj−1 djj dj+1 · · · dar r
,
where we used the fact that the Dirichlet series ∞ X
G(s1 , . . . , sr ) =
n1 ,...,nr
g(n1 , . . . , nr ) ns11 · · · nsrr =1
is absolutely convergent when ℜ(s1 + · · · + sr ) > a1 + · · · + ar − δ. Now we evaluate the sum r X Y x . M (x) := Mj g(d1 , . . . , dr ) dj j=1
d1 ,...,dr ≤x
Since Mj (u) = xaj Pj (u) with Pj (u) a polynomial in u of degree δj , we have r Y
j=1
Mj
x dj
=
xa1 +···+ar da11 · · · dar r
where Cℓ (log d1 , . . . , log dr ) =
δ1 +···+δ X r
Cℓ (log d1 , . . . , log dr )(log x)ℓ ,
ℓ=0
X
c(j1 , . . . , jr )(log d1 )j1 · · · (log dr )jr ,
j1 ,...,jr
the sum being over 0 ≤ jt ≤ δt (1 ≤ t ≤ r). So we have M (x)
= xa1 +···+ar
δ1 +···+δ X r
d1 ,...,dr ≤x
ℓ=0
= xa1 +···+ar
δ1 +···+δ X r
+ xa1 +···+ar
dℓ :=
ℓ=0 δ1 +···+δ X r
∞ X
g(d1 , . . . , dr )Cℓ (log d1 , . . . , log dr ) da11 · · · dar r
(17)
dℓ (log x)ℓ
ℓ=0
where
X
(log x)ℓ
d1 ,...,dr =1
(log x)ℓ
X′ g(d1 , . . . , dr )Cℓ (log d1 , . . . , log dr ) , da11 · · · dar r
d1 ,...,dr
g(d1 , . . . , dr )Cℓ (log d1 , . . . , log dr ) da11 · · · dar r
P′
means that there is at least one j (1 ≤ j ≤ r) such that dj > x. Without loss of and where generality, we suppose dr > x. 13
Suppose ε > 0 is sufficiently small and we have log n ≪ nε . Thus xa1 +···+ar
δ1 +···+δ X r
(log x)ℓ
ℓ=0
≪x
a1 +···+ar
X′ g(d1 , . . . , dr )Cℓ (log d1 , . . . , log dr ) da11 · · · dar r
d1 ,...,dr
δ1 +···+δ X r
ℓ
(log x)
ℓ=0
≪x
a1 +···+ar
δ1 +···+δ X r
ℓ
(log x)
≪x
X |g(d1 , . . . , dr )|dδ1 ε · · · dδr ε r 1 da11 · · · dar r
∞ X
X |g(d1 , . . . , dr )|dδ1 ε · · · dδr ε r 1
d1 ,...,dr−1 =1 dr >x δ1 +···+δ X r ℓ=0
a1 +···+ar −(ar −br )
∞ X
d1 ,...,dr−1 =1 dr >x
ℓ=0
≪ xa1 +···+ar −(ar −br )
(18)
(log x)ℓ
da11
ar−1 br · · · dr−1 dr
×
1 drar −br
∞ X
|g(d1 , . . . , dr )| ar−1 −δr−1 ε br −δr ε a1 −δ1 ε · · · dr−1 dr d1 ,...,dr =1 d1
(log x)δ1 +···+δr ≪ xa1 +···+ar −∆1 (log x)δ1 +···+δr .
From (17) and (18) we get M (x) = xa1 +···+ar
δ1 +···+δ X r
dℓ (log x)ℓ + O(xa1 +···+ar −∆1 (log x)δ1 +···+δr ).
(19)
ℓ=0
Now Theorem 3.3 follows from (14), (15), (16) and (19). Proof of Theorem 3.4. Apply Theorem 3.3 in the case fj (n) = τ (n), aj = 1, bj = θ + ε, (1 ≤ j ≤ r) by using Proposition 2.1. Proof of Theorem 3.5. Apply Theorem 3.3 in the case fj (n) = τ ∗ (n), aj = 1, bj = 1/2, (1 ≤ j ≤ r) by using Proposition 2.5.
References [1] S. W. Graham and G. Kolesnik, On the difference between consecutive squarefree integers, Acta Arith. 49 (1988), 435–447. [2] T. Hilberdink and L. T´ oth, On the average value of the least common multiple of k positive integers, J. Number Theory 169 (2016), 327–341. [3] M. N. Huxley, Exponential sums and lattice points III., Proc. London Math. Soc. 87 (2003), 591–609. [4] E. Kr¨ atzel, Lattice points, Kluwer, Dordrecht–Boston–London, 1988. [5] A. V. Lelechenko, Average number of squares dividing mn, Visn. Odessk. Univ., Ser. Mat. Mekh. 19 #2 (22) (2014), 52–65. [6] D. Suryanarayana and V. Siva Rama Prasad, The number of k-free divisors of an integer, Acta Arith. 17 (1971), 345–354.
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[7] L. T´ oth, Two generalizations of the Busche-Ramanujan identities, Int. J. Number Theory 9 (2013), 1301–1311. [8] L. T´ oth, Multiplicative arithmetic functions of several variables: a survey, in Mathematics Without Boundaries, Surveys in Pure Mathematics. Th. M. Rassias, P. Pardalos (Eds.), Springer, New York, 2014, 483–514. [9] L. T´ oth, Counting r-tuples of positive integers with k-wise relatively prime components, J. Number Theory 166 (2016), 105–116. [10] J. Wu, Probl`eme de diviseurs exponentiels et entiers exponentiellement sans facteur carr´e, J. Th´eor. Nombres Bordeaux 7 (1995), 133–141. L´ aszl´ o T´ oth Department of Mathematics University of P´ecs Ifj´ us´ ag u ´tja 6, H-7624 P´ecs, Hungary E-mail:
[email protected] Wenguang Zhai Department of Mathematics China University of Mining and Technology Beijing 100083, China E-mail:
[email protected]
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