Abstract
Let \(f_{3}\) denote the characteristic function of cube-full numbers, and let (n, q) be the greatest common divisor of positive integers n and q. For any positive real numbers x and y, we shall consider several asymptotic formulas for sums of sums of modified cube-full numbers, which is \( \sum _{n\le y}\left( \sum _{q\le x}\sum _{d|(n,q)}df_{3}\left( {q}/{d}\right) \right) ^{k} \) with \(k=1, 2\).
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1 Introduction
For any integer \(r (\ge 2)\), we call n an r-full integer if \(p|n \Rightarrow p^{r}|n\) and call n an r-free integer if \(p|n \Rightarrow p^{r}\not \mid n\), where the letter p denotes a prime number. If \(r=2\) or \(r=3\), we use the terms square-full or cube-full. Let G(r) denote the set of r-full numbers, then we set
Let \(s=\sigma +it\) be the complex variable, and let \(\zeta (s)\) be the Riemann zeta-function. Denote the Dirichlet series \(F_{r}(s)\) defined by \( F_{r}(s) := \sum _{n=1}^{\infty }\frac{f_{r}(n)}{n^s}. \) Following (7.3) in Krätzel [7], the representation of \(F_{r}(s)\) is more complicated for \(k\ge 3\), and it is known that
holds. Here, \(c_{2r+3}(n)\) denotes a certain arithmetical function whose associated Dirichlet series \( \kappa _{2r+3}(s) := \sum _{n=1}^{\infty }\frac{c_{2r+3}(n)}{n^s}, \) which is absolutely convergent for \(\mathrm{Re}~s >\frac{1}{2r+3}\). We define a sum over the r-full numbers by
where (n, q) denotes the greatest common divisor of integers n and q. For any large positive real numbers x and y, we set the double sums
For \(r=2\), Kiuchi [6] considered the asymptotic formula for the double sum (1.3) concerning square-full numbers, and used the theory of exponent pairs to derive the precise asymptotic formula
where x and y are large real numbers such that \(x \ll y \ll x^{\frac{3}{2}}\). When \(k=2\), Kiuchi [6] also showed that
holds, where x and y are large real numbers such that \( y \gg \frac{x^2}{\log x}\). Moreover, he used analytic properties of the Riemann zeta-function to obtain the asymptotic formula
where \(c_0\) is a computable constant, and x and y are large real numbers such that \(x \log ^{4}x \ll y \ll \frac{x^2}{\log ^{6}x}\). To prove the precise asymptotic formulas (1.4), (1.5) and (1.6), we used the method of proofs of Chan and Kumchev [2] (see also Kiuchi, Minamide and Tanigawa [5], Kühn and Robles [8], Robles [10], Robles and Roy [11]).
For \(r=3\), it is derived from (1.1) that the Dirichlet series for the generating function \(f_{3}(n)\) is
for \(\mathrm{{Re}}~s > \frac{1}{3}\), where \(\kappa (s)\) is the Dirichlet series generated by a certain arithmetical function \(c_{9}(n)\) (see (7.3) in Krätzel [7]), that is \( \kappa (s):=\sum _{n=1}^{\infty }\frac{c_{9}(n)}{n^s} \) which is absolutely convergent for \(\mathrm{Re}~s > \frac{1}{9}\). Moreover, the asymptotic formula for the sum of \(f_{3}(n)\) is also known, and one can see that
holds with the error term \( \Delta (x) = O\left( x^{\frac{1}{8}}\log ^{4}x\right) \) for any large positive real number x (see section 7.1.3 in Krätzel [7]). In 1988, Balasubramanian et al. [1] showed that \( \Delta (x)=\Omega \left( x^{\frac{1}{12}}\sqrt{\log x}\right) \) holds, and the improvement on the estimate of \(\Delta (x)\) has been studied by many authors. Under the Riemann hypothesis, Wu [14] obtained that \( \Delta (x)=O\left( x^{\frac{97}{804}+\varepsilon }\right) \) holds for any \(\varepsilon >0\). Using (1.7), the Dirichlet series generated by the coefficients \(s_{q}(n)\) is expressed by
for \({\mathrm{{Re}}~{s} > \frac{1}{3}}\), where \(\sigma _{1-s}(n)=\sum _{d|n}d^{1-s}\) is the generalized divisor function.
Now, we shall consider several asymptotic formulas of (1.3) concerning cube-full numbers. Our theorems are proved by the same way as in [6], and we shall deduce several interesting formulas for the double sum \(S_{k}^{(3)}(x,y)\). We use the theory of exponent pairs and elementary methods to deal with \(S_{1}^{(3)}(x,y)\). Then the case \(k=1\) implies the following theorem, namely
Theorem 1
Let x and y be large real numbers such that \(x \ll y \ll x^{\frac{5}{3}}\). Then we have
It follows from (1.10) that
holds. This is described by saying that the average order of \(s_{q}^{(3)}(n)\) is \( \frac{\zeta (3)\zeta (4)\zeta (5)\kappa (1)}{\zeta (8)} \) under q and n satisfying the condition \( q^{} \ll n \ll q^{\frac{5}{3}}. \)
Remark 1.1
It would be an interesting problem to investigate the asymptotic behaviour of \(S_{1}^{(3)}(x,y)\) under the condition \( y \ll x^{}\). However, this would require a different method.
For \(k=2\), there are two quite different methods to deal with this function \(S_{2}^{(3)}(x,y)\). We utilize an elementary lattice point counting argument to obtain the formula (1.11) below, and use the generating Dirichlet series and the properties of the Riemann zeta-function to prove (1.12) below, which we state as
Theorem 2
Let x and y be large real numbers such that \( y \gg \frac{x^2}{\log x}\). Then we have
Similarly, as in Theorem 1, we use (1.11) to get
This is described by saying that the average order of \(s_{q}^{(3)}(n)\) is
under q and n satisfying the condition \( n \gg \frac{q^2}{\log q}. \) We utilize the generating Dirichlet series and the properties of the Riemann zeta-function to prove (1.12) below, which we state as
Theorem 3
Let x and y be large real numbers such that \(x \log ^{6}x \ll y \ll \frac{x^2}{\log ^{4}x}\). Then we have
where \(\eta \) is a computable constant, which is defined by (5.9) below, and the constant \(c_{1}\) is given by
Remark 1.2
It would be an interesting problem to investigate the asymptotic behaviour of \(S_{2}^{(3)}(x,y)\) under the condition \( y \ll x \log ^{6}x\). However, this would require a different method.
2 Some lemmas
To prove our theorems, we first prepare several lemmas. Let \(\psi (x)=x-[x]-\frac{1}{2}\) denote the first periodic Bernoulli function. In the proof of Theorem 1, we need an upper bound of the sum
An efficient way to estimate these \(\psi \)-sums is to apply the theory of exponent pairs: An exponent pair \((\kappa , \lambda )\) is a pair of numbers \(0 \le \kappa \le \frac{1}{2} \le \lambda \le 1\) such that
holds, where \(I \subset (N,2N]\) and \(A \ll |f'(u)| \ll A\) for \(u \in I\). For the precise definition and its properties, the reader should consult Graham and Kolesnik [3] and Ivić [4]. Now applying Lemma 4.3 in [3] with \(f(n)=\frac{y}{n}\), we have
Lemma 2.1
Let \((\kappa ,\lambda )\) be an exponent pair. If I is a subinterval in (N, 2N], we have
In particular, if we take the exponent pair \((\kappa , \lambda )=\left( \frac{1}{2}, \frac{1}{2}\right) \), we get
The proofs of Theorem 3 need the following lemmas, namely
Lemma 2.2
Suppose that the Dirichlet series \( \alpha (s):=\sum _{n=1}^{\infty }\frac{a_n}{n^s} \) absolutely converges for \(\mathrm{Re}~s >\sigma _{a}\). If \(\sigma _{0}>\max (0,\sigma _{a})\) and \(x>0,\ T>0\), then
where
and \(\sum {}^{'}\) indicates that the last term is to be halved if x is an integer.
Proof
This is Perron’s famous formula (see Theorem 5.2 and Corollary 5.3 in Montgomery and Vaughan [9]). \(\square \)
Lemma 2.3
Let \(G(s_1,s_2;y)\) be a sum function defined by
and \(L=\log y\). Then we have
for \(\mathrm{{Re}}~s_{j} \ge 1/2\) and \(|\mathrm{{Im}}~s_{j}| \le T \ (j=1,2)\), where
where \(\sum {}^{'}\) indicates that the last term is to be halved if y is an integer.
Proof
The proof of this lemma follows from (4.12) in Chan and Kumchev [2]. \(\square \)
Lemma 2.4
For \(t\ge t_{0} >0\) uniformly in \(\sigma \), we have
Proof
The proof of this lemma follows from Theorem II.3.8 in Tenenbaum [12], and Ivić [4]. Also see Titchmarsh [13]. \(\square \)
3 Proof of Theorem 1
We use (1.2) and (1.3) and change the order of summation to obtain
We consider the first term on the right of (3.1). We use (1.7) to get
We obtain from (1.7) and the above
Similarly, we have
To estimate \(S_{1,3}(x,y)\), we use the theory of exponent pairs. Let \(\displaystyle N_{j}=N_{j,k}=\left( \frac{x}{k}\right) 2^{-j}\). Then we have
where the \(\sup \) is over all subintervals I in \((N_{j},2N_{j}]\). From (2.1) of Lemma 2.1 and (3.2), we have
Substituting (3.3), (3.4) and (3.5) into (3.1), we get the assertion of Theorem 1.
\(\square \)
4 Proof of Theorem 2
where \([d_1,d_2]\) denotes the least common multiple of \(d_1\) and \(d_2\). We use (1.7) to get
To evaluate the main term of (4.1), we use
which follows from (1.7) and (3.2). Using the Gauss identity \( \sum _{d|n}\phi (d) = n, \) (4.2) and \( \sum _{d\le x}\frac{\phi (d)}{d^2}=\frac{1}{\zeta (2)}\log x +O(1) \), we have
Hence, we have
This completes the proof of Theorem 2. \(\square \)
5 Proof of Theorem 3
In this section, we assume that \(1 \le y \le x^M\) for some constant M. Without loss of generality we can assume that \(x,\ y\in {\mathbb {Z}}+\frac{1}{2}\). We apply Lemma 2.2 with (1.9), then
with \(\alpha = 1 +\frac{1}{\log x}\) and T being a real parameter at our disposal, where \(E_{1}(x,n)\) is the error term given by
by using (1.9). Let \(\alpha _{1}=1 + \frac{1}{\log x}\) and \(\alpha _{2}=1 + \frac{2}{\log x}\). Applying (5.1) with \(\alpha =\alpha _j\ (j=1,2)\) we have
where
and
It follows that
Summing (5.2) over n and using the estimate \( \sum _{n\le y}\sigma _{0}(n)^{2} \ll y^{}\log ^{3}y, \) we get
where \( G(s_1,s_2;y) := \sum _{n\le y}\sigma _{1-s_{1}}(n)\sigma _{1-s_{2}}(n) \) and \(L=\log (Txy)\).
Now we shall evaluate the integrals in appearing in (5.3). Substituting (2.3) into (5.3), we have
where
Note that we substitute \(T=x^{}\) into the error term on the right-hand side of (5.4) to get
5.1 Evaluation of \(S_{2,1}^{(3)}(x,y)\)
Let \(\alpha _{1}= 1 +\frac{1}{\log x}\) and \(\alpha _{2}= 1 + \frac{2}{\log x}\). From the definition of \(R_1(s_1,s_2,y)\), we get
Let \(\Gamma (\alpha , \beta ,T)\) denote the contour consisting of the line segments \([\alpha -iT, \beta -iT]\), \([\beta -iT, \beta +iT]\) and \([\beta +iT, \alpha +iT]\). In (5.6), we move the integration with respect to \(s_2\) to \(\Gamma (\alpha _2, \frac{1}{2}+\frac{1}{\log x},T)\). We denote the integrals over the horizontal line segments by \(J_{1,1}\) and \(J_{1,3}\), and the integral over the vertical line segment by \(J_{1,2}\), respectively. Then using the estimate \(\int _1^T |\zeta (\alpha _1+it)|dt \ll T\) and Lemma 2.4, we have
For the integral along the vertical line we have
Hence we use the estimate
(see p.161 in [5]) and the Cauchy–Schwarz inequality to get
It remains to evaluate the residues of the poles of the integrand when we move the line of integration to \(\Gamma (\alpha _2,\frac{1}{2}+\frac{1}{\log x},T)\). There exists a simple pole at \(s_2=2-s_1\) with residue
and also a simple pole at \(s_2=1\) with residue
The contributions to \(S_{2,1}^{(3)}(x,y)\) from these residues are
For \(I_1\), moving the line of integration to \(\Gamma (\alpha _1, \frac{5}{4},T)\), we have
where the constant \(\eta \) is given by
which is an absolutely convergent integral given by
Now, we use the inequalities \( |\zeta (s)| \le \zeta (\sigma ) \) and \( \left| \frac{1}{\zeta (s)}\right| \le \frac{\zeta (\sigma )}{\zeta (2\sigma )} \) for \(\sigma >1\) (see (8.4.1), (8.7.1) in [13]) to obtain
Here, the integral on the right-hand side of the above is a computable constant, and that is, strictly speaking, enough for the purpose of this paper.
For \(I_2\), we move the line of integration to \(\Gamma (\alpha _1,\frac{1}{2}+\frac{1}{\log x},T)\). The integrals over the horizontal lines are
and the integral over the vertical line is
by using the estimate \(\int _{1}^{T}|\zeta (\frac{1}{2}+it)|^{2}dt \ll T\log T\) and integration by parts. Furthermore, when moving the path of integration there is a double pole at \(s_1=1\). Hence, using Cauchy’s theorem, we have
where \(\gamma \) is the Euler constant. Combining these results we have
Here, we substituted \(T=x^{}\) into the error term of \(S_{2,1}(x,y)\).
5.2 Estimation of \(S_{2,4}^{(3)}(x,y)\)
Explicitly we have
For this purpose, we move the line of integral with respect to \(s_2\) to contour \(\Gamma (\alpha _{2}, \beta , T)\), where \(\beta =\frac{5}{2} -\alpha _{1}=\frac{3}{2} - \frac{1}{\log x}\). We denote the integrals over the horizontal line segments by \(J_{4,1}\) and \(J_{4,3}\), and the integral over the vertical line segment by \(J_{4,2}\), respectively. There are no poles when we deform the path of integral over \(s_2\). The contribution from the horizontal lines are
The inner integral is estimated as
where we have used the assumption \(y \ll x^M\). Hence, we have
For the integral on the vertical line we find that
Hence, we take \(T=x^{}\) to get
5.3 Estimation of \(S_{2,3}^{(3)}(x,y)\)
It is given explicitly by
We move the path of integration with respect to \(s_2\) to \(\Gamma (\alpha _2, \frac{3}{2}, T)\). We denote the integrals over the horizontal line segments by \(J_{3,1}\) and \(J_{3,3}\), and the integral over the vertical line segment by \(J_{3,2}\), respectively. Note that there exist no poles with this deformation. The contribution from the horizontal lines are
On the other hand, the contribution from the vertical lines is
Hence, we take \(T=x^{}\) into the above to obtain
5.4 Evaluation of \(S_{2,2}^{(3)}(x,y)\)
The explicit form of \(S_{2,2}^{(3)}(x,y)\) is given by
This time we firstly move the line of the integration over \(s_1\) to \(\Gamma (\alpha _{1},\frac{3}{2},T)\). The estimates over the horizontal lines and the vertical line are the same as that of \(S_{2,3}^{(3)}(x,y)\), but there is a simple pole at \(s_1=s_2\) inside this contour. The residue of the integrand of (5.13) at this pole is
Hence, we have
by taking \(T=x^{}\). We move the line of integration to \(\Gamma (\alpha _2,\frac{1}{2}+\frac{2}{\log x},T)\). By the same method as before, the integrals over the horizontal lines are estimated as
and the vertical lines are estimated as
Furthermore, there is a contribution from the pole \(s_2=1\) of order 2, hence \(S_{2,2}^{(3)}(x,y)\) has the form
by taking \(T=x^{}\).
5.5 Asymptotic formula of (1.12)
Now, we substitute (5.5), (5.10), (5.11), (5.12) and (5.14) into (5.4) to obtain the assertion of Theorem 3. \(\square \)
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Kiuchi, I. On sums of sums involving cube-full numbers. Ramanujan J 59, 279–296 (2022). https://doi.org/10.1007/s11139-021-00524-6
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DOI: https://doi.org/10.1007/s11139-021-00524-6