On the reciprocal sum of lcm of $k$-tuples

Sungjin Kim
---
Santa Monica College, California State University Northridge
<EMAIL_ADDRESS>

###### Abstract

We prove that the reciprocal sum $S_{k}(x)$ of the least common multiple of
$k\geq 3$ positive integers in $\mathbb{N}\cap[1,x]$ satisfies

$S_{k}(x)=P_{2^{k}-1}(\log x)+O(x^{-\theta_{k}+\epsilon})$

where $P$ is a polynomial of degree $2^{k}-1$ and
$\theta_{k}=\frac{2^{k}}{(k+1)^{\frac{k+1}{2}}}\cdot\frac{3}{2^{k}+6k-5}$.
This was conjectured in Hilberdink, Luca, and Tóth [HLT, Remark 2.4]. We also
prove asymptotic formulas for similar sums conjectured there.

##  1 Introduction

Let $x\geq 2$ be an integer, $\mathbb{N}=\\{1,2,3,\ldots\\}$ be the set of
positive integers, and $\mathcal{P}$ be the set of prime numbers. Let
$(n_{1},\ldots,n_{k})=\gcd(n_{1},\ldots,n_{k})$ and
$[n_{1},\ldots,n_{k}]=\textrm{lcm}(n_{1},\ldots,n_{k})$ be the greatest common
divisor (gcd) and the least common multiple (lcm) of $k$-tuple of positive
integers $n_{1},\ldots,n_{k}$ respectively. If $(n_{1},\ldots,n_{k})$ means
the $k$-tuple of positive integers, then they will be written as
$(n_{1},\ldots,n_{k})\in\mathbb{N}^{k}$. In this paper, we denote $P_{v}(x)$,
$P^{(1)}_{v}(x)$, $P^{(2)}_{v}(x)$, $\ldots$ , polynomials of degree at most
$v$ in variable $x$. They are not necessarily the same at each occurrence.

The reciprocal sum of gcd is well understood through Dirichlet convolution of
arithmetic functions. We have [HLT, Page 3] for $k\geq 3$,

$\sum_{n_{1},\ldots,n_{k}\leq
x}\frac{1}{(n_{1},\ldots,n_{k})}=\frac{\zeta(k+1)}{\zeta(k)}x^{k}+O(x^{k-1}),$

and by [L, Theorem 1.1], we have for $k=2$,

$\sum_{n_{1},n_{2}\leq
x}\frac{1}{(n_{1},n_{2})}=\frac{\zeta(3)}{\zeta(2)}x^{2}+O\left(x(\log
x)^{2/3}(\log\log x)^{1/3}\right).$

The reciprocal sums involving the lcm are more delicate. The $r$-th moment
results were studied by Hilberdink and Tóth [HT] by means of multivariable
Dirichlet series. They proved that for $r>-1$,

$\sum_{n_{1},\ldots,n_{k}\leq
x}[n_{1},\ldots,n_{k}]^{r}=\frac{C_{r,k}}{(r+1)^{k}}x^{k(r+1)}+O\left(x^{k(r+1)-\frac{1}{2}\textrm{min}(r+1,1)+\epsilon}\right)$
(1)

and

$\sum_{n_{1},\ldots,n_{k}\leq x}\left(\frac{[n_{1},\ldots,n_{k}]}{n_{1}\cdots
n_{k}}\right)^{r}=C_{r,k}x^{k}+O\left(x^{k-\frac{1}{2}\textrm{min}(r+1,1)+\epsilon}\right)$
(2)

where $C_{r,k}$ is given as an Euler product.

The results in case $r=-1$ in (1) and (2) were not obtained in [HT].
Hilberdink, Luca, and Tóth [HLT] studied these sums and proposed some open
problems. They proved that as $x\rightarrow\infty$,

$S_{k}(x):=\sum_{n_{1},\ldots,n_{k}\leq
x}\frac{1}{[n_{1},\ldots,n_{k}]}\asymp\log^{2^{k}-1}x,$ (3)
$U_{k}(x):=\sum_{\begin{subarray}{c}{n_{1},\ldots,n_{k}\leq x}\\\
{(n_{1},\ldots,n_{k})=1}\end{subarray}}\frac{1}{[n_{1},\ldots,n_{k}]}\asymp\log^{2^{k}-2}x,$
(4)

and

$x^{k}\ll V_{k}(x):=\sum_{n_{1},\ldots,n_{k}\leq x}\frac{n_{1}\cdots
n_{k}}{[n_{1},\ldots,n_{k}]}\ll x^{k}(\log x)^{2^{k}-2}.$ (5)

Then they conjectured that

$S_{k}(x)=P_{2^{k}-1}(\log x)+O(x^{-r})$

and

$V_{k}(x)\sim\lambda_{k}x^{k}(\log x)^{2^{k}-k-1},$

where $P_{2^{k}-1}$ is a polynomial of degree $2^{k}-1$ with a positive
leading coefficient, $r>0$, and $\lambda_{k}>0$.

Recently, the author [K, Theorem 1.2, and 1.3] improved the error terms of (1)
and (2) and provided the correct order of magnitude of (5). In fact,

$\sum_{n_{1},\ldots,n_{k}\leq
x}[n_{1},\ldots,n_{k}]^{r}=\frac{C_{r,k}}{(r+1)^{k}}x^{k(r+1)}+O\left(x^{k(r+1)}E_{r,k}(x)\right)$

and

$\sum_{n_{1},\ldots,n_{k}\leq x}\left(\frac{[n_{1},\ldots,n_{k}]}{n_{1}\cdots
n_{k}}\right)^{r}=C_{r,k}x^{k}+O\left(x^{k}E_{r,k}(x)\right)$

where

$E_{r,k}(x)=\begin{cases}O_{r}(x^{-\frac{r+1}{2}}\log^{\max(2^{k}-k-1,2k^{2}-k-2)}x)\mbox{
if }-1<r\leq 1,\\\ O_{r}(x^{-1}\log^{k-1}x)\mbox{ if }r>1.\end{cases}.$

Moreover,

$V_{k}(x)\asymp x^{k}(\log x)^{2^{k}-k-1}.$

The method in this recent work was based on the study of $G$-wise coprime
tuples by Tóth [To] and Hu [H1], [H2]. The case when $G=K_{k}$ is a complete
graph was sufficient.

L. Tóth informed the author that D. Essouabri, C. Salinas Zavala, L. Tóth
[EST, Corollary 2, 3, and 4] proved among other things that there exist
$\theta>0$ such that

$S_{k}(x)=\sum_{n_{1},\ldots,n_{k}\leq
x}\frac{1}{[n_{1},\ldots,n_{k}]}=P^{(1)}_{2^{k}-1}(\log x)+O(x^{-\theta}),$
$U_{k}(x)=\sum_{\begin{subarray}{c}{n_{1},\ldots,n_{k}\leq x}\\\
{(n_{1},\ldots,n_{k})=1}\end{subarray}}\frac{1}{[n_{1},\ldots,n_{k}]}=P^{(2)}_{2^{k}-2}(\log
x)+O(x^{-\theta}),$

and

$V_{k}(x)=\sum_{n_{1},\ldots,n_{k}\leq x}\frac{n_{1}\cdots
n_{k}}{[n_{1},\ldots,n_{k}]}=x^{k}P^{(3)}_{2^{k}-k-1}(\log
x)+O(x^{k-\theta}).$

The leading coefficients of $P_{2^{k}-1}^{(1)}$, $P_{2^{k}-2}^{(2)}$, and
$P_{2^{k}-k-1}^{(3)}$ are explicitly determined in [EST, Corollary 2, 3, and
4].

In this paper, we prove explicit power-savings in these estimates. The leading
coefficients of $P_{2^{k}-1}^{(1)}$, $P_{2^{k}-2}^{(2)}$, and
$P_{2^{k}-k-1}^{(3)}$ are determined in Section 3 and 4 independently of
[EST].

######  Theorem 1.1.

As $x\rightarrow\infty$, we have for $k\geq 3$,

$S_{k}(x)=P^{(1)}_{2^{k}-1}(\log x)+O(x^{-\theta^{(1)}_{k}+\epsilon}),$ (6)
$U_{k}(x)=P^{(2)}_{2^{k}-2}(\log x)+O(x^{-\theta^{(2)}_{k}+\epsilon}),$ (7)
$V_{k}(x)=x^{k}P^{(3)}_{2^{k}-k-1}(\log x)+O(x^{k-\theta^{(3)}_{k}+\epsilon})$
(8)

where the leading coefficient of $P_{2^{k}-1}^{(1)}$ is $c_{k}^{(1)}=c_{k}>0$,
the leading coefficient $c_{k}^{(2)}$ of $P_{2^{k}-2}^{(2)}$ satisfies
$c_{k}^{(2)}=(2^{k}-1)c_{k}$. For each $k\geq 2$, $c_{k}>c_{k+1}$ and
$\lim_{k\rightarrow\infty}c_{k}=0$. Moreover, for any $1\leq i<j\leq 3$, we
have $c_{k}^{(i)}/c_{k}^{(j)}\in\mathbb{Q}\cap(0,\infty)$. We have for $k\geq
3$,

$\theta^{(1)}_{k}=\theta^{(3)}_{k}=\frac{2^{k}}{(k+1)^{\frac{k+1}{2}}}\cdot\frac{3}{2^{k}+6k-5},\
\
\theta^{(2)}_{k}=\frac{2^{k}}{(k+1)^{\frac{k+1}{2}}}\cdot\frac{3}{2^{k}+6k-6}.$

Our method relies on the non-inductive argument in the counting of $G$-wise
coprime tuples by Arias-de-Reyna and Heyman [AH]. By their argument, we are
able to write $S_{k}(x)$ as a sum over $(2^{k}-1)$-tuples of positive
integers. We prove a general result (Theorem 2.2) on the reciprocal sum over
$G$-wise coprime tuples in Section 2 by an elementary method, then we obtain
the leading terms of $S_{k}(x)$, $U_{k}(x)$, and $V_{k}(x)$ as a corollary in
Section 3. Further in Section 3, we apply a version of multivariable Perron’s
formula to strengthen Theorem 2.2 into a full asymptotic formula with a power-
saving error term. Then we complete the proof of Theorem 1.1 as a corollary.
The computations of $c_{k}$ are obtained in Section 4. All asymptotic formulas
given in this paper are for $x\rightarrow\infty$. The notations $A(x)\ll B(x)$
or $A(x)=O(B(x))$ mean that $|A(x)/B(x)|$ is bounded as $x\rightarrow\infty$.
For simplicity of exposition, the positive numbers $M$ without subscripts and
$\epsilon$ may appear multiple times, but they are not necessarily the same
everytime. We use $\mathcal{L}=\log x$ and $\mathcal{L}^{M}$ for any fixed
power of $\log x$ so that $O(\mathcal{L}^{M})\cdot
O(\mathcal{L}^{M})=O(\mathcal{L}^{M})$, any divisor function
$\tau_{v}(n)=\sum_{a_{1}\cdots a_{v}=n}1$ satisfies $\tau_{v}(n)\ll
n^{\epsilon}$, $\mathcal{L}^{M}\ll x^{\epsilon}$, and $O(x^{\epsilon})\cdot
O(x^{\epsilon})=O(x^{\epsilon})$.

Data Availability Statement. All data generated or analysed during this study
are included in this published article and its supplementary information
files.

Acknowledgments. The author thanks Titus Hilberdink for initially mentioning
the multivariable Perron’s formula. The author also thanks Sary Drappeau for
bringing [dlB, Theorem 1] to his attention and helpful conversation about the
paper.

##  2 $G$-wise coprime $k$-tuples

Let $x\geq 2$ be an integer and $G=(V,E)$ be any graph with $v\geq 2$ vertices
and $e\geq 1$ edges. Let $V=\\{1,\ldots,v\\}$ and
$E=\\{\epsilon_{1},\ldots,\epsilon_{e}\\}\subseteq\\{(i,j)\ |\ 1\leq i<j\leq
v\\}$ so that we have labels on vertices and edges of $G$. We are interested
in a reciprocal sum over $v$-tuples
$(a_{1},\ldots,a_{v})\in(\mathbb{N}\cap[1,x])^{v}$ such that $(a_{i},a_{j})=1$
whenever $(i,j)\in E$. Such $v$-tuples are called $G$-wise coprime in [H1],
[H2], and [AH]. The asymptotic formula of the counting function of $G$-wise
coprime tuples is obtained in [H2] by an inductive argument and [AH] by an
non-inductive argument. For each vertex $r$, let $M_{r}$ be the least common
multiple of $m_{i}$ where the edge $\epsilon_{i}$ connects the vertex $r$ to
another vertex of $G$. The number of $G$-wise coprime tuples given in [AH] is

$\sum_{\begin{subarray}{c}{a_{1},\ldots,a_{v}\leq x}\\\ {G\textrm{-wise
coprime}}\end{subarray}}1=\sum_{m_{1},\ldots,m_{e}\leq
x}\mu(m_{1})\cdots\mu(m_{e})\prod_{r=1}^{v}\left\lfloor\frac{x}{M_{r}}\right\rfloor.$

They obtained this formula by inclusion-exclusion principle. We provide a more
direct approach to prove this. Recall that when we sum over the coprime pairs
$(a,b)$, we insert $\sum_{d|a,d|b}\mu(d)$ into the summation. To each edge
$\epsilon_{i}=(a,b)\in E$, we insert $\mu(m_{i})$ and the divisibility
conditions $m_{i}|a$, $m_{i}|b$. Combining all divisibility conditions of each
vertex $r$, we obtain the result. Similarly for reciprocal sums, we have

$\sum_{\begin{subarray}{c}{a_{1},\ldots,a_{v}\leq x}\\\ {G\textrm{-wise
coprime}}\end{subarray}}\frac{1}{a_{1}\cdots
a_{v}}=\sum_{m_{1},\ldots,m_{e}\leq
x}\mu(m_{1})\cdots\mu(m_{e})\sum_{\begin{subarray}{c}{a_{1},\ldots,a_{v}\leq
x}\\\ {\forall_{r},M_{r}|a_{r}}\end{subarray}}\frac{1}{a_{1}\cdots a_{v}}.$
(9)

The asymptotic formula for the reciprocal sums over $G$-wise coprime tuples is
as follows.

######  Theorem 2.1.

Let $d$ be the maximal degree of the vertices of $G$. We have

$\sum_{\begin{subarray}{c}{a_{1},\ldots,a_{v}\leq x}\\\ {G\textrm{-wise
coprime}}\end{subarray}}\frac{1}{a_{1}\cdots a_{v}}=P_{v}(\log
x)+O(x^{-1}\log^{v+d-1}x)$

where $P_{v}$ is a polynomial of degree $v$ with a leading coefficient
$\rho(G)>0$.

###### Proof.

The inner sum of (9) is

$\frac{1}{M_{1}\cdots
M_{v}}\sum_{\forall_{r},b_{r}\leq\frac{x}{M_{r}}}\frac{1}{b_{1}\cdots
b_{v}}=\frac{1}{M_{1}\cdots
M_{v}}\prod_{r=1}^{v}\left(\log\frac{x}{M_{r}}+\gamma+O\left(\frac{M_{r}}{x}\right)\right).$

It was proved in [AH] that for any $1\leq r\leq v$ with obvious modification
for $r=1,v$,

$\sum_{m_{1},\ldots,m_{e}\leq
x}|\mu(m_{1})\cdots\mu(m_{e})|\frac{1}{M_{1}\cdots M_{r-1}M_{r+1}\cdots
M_{v}}=O(\log^{d}x)$

Thus, the $O$-terms in the product contribute $O(x^{-1}\log^{v-1+d}x)$.

The coefficients of $P_{v}$ are linear combinations of the absolutely
convergent sums

$\sum_{m_{1},\ldots,m_{e}=1}^{\infty}\mu(m_{1})\cdots\mu(m_{e})\frac{(\gamma-\log
M_{r_{1}})\cdots(\gamma-\log M_{r_{s}})}{M_{1}\cdots M_{v}}$

and the sums over $m_{i}>x$ for some $i\leq e$ in the above contribute to the
error term. In fact, it was proved in [AH] that if $m_{i}$ is the largest
among $m_{1},\ldots,m_{e}$, then

$\sum_{m_{1},\ldots,m_{i-1}\leq
m_{i}}\sum_{m_{i}>x}\sum_{m_{i+1},\ldots,m_{e}\leq
m_{i}}|\mu(m_{1})\cdots\mu(m_{e})|\frac{1}{M_{1}\cdots
M_{v}}=O\left(\frac{(\log\log x)^{w}}{x}\right)$ (10)

for some $w=w(G)>0$. Applying the absolute convergence of the Dirichlet series
of
$\zeta(s)^{-1}\sum\frac{|\mu(n)|}{n^{s}}\left(\frac{\sigma(n)}{n}\right)^{w}$
over $\Re(s)>1/2$, we have

$\sum_{n>x}\frac{|\mu(n)|}{n^{2}}\left(\frac{\sigma(n)}{n}\right)^{w}=O\left(\frac{1}{x}\right).$

Thus, the error term of (10) can be improved to $O\left(\frac{1}{x}\right)$.
This yields

$\displaystyle\sum_{m_{1},\ldots,m_{i-1}\leq m_{i}}$
$\displaystyle\sum_{m_{i}>x}\sum_{m_{i+1},\ldots,m_{e}\leq
m_{i}}|\mu(m_{1})\cdots\mu(m_{e})|\frac{(\gamma-\log
M_{r_{1}})\cdots(\gamma-\log M_{r_{s}})}{M_{1}\cdots
M_{v}}=O\left(\frac{\log^{s}x}{x}\right).$

The error term from the coefficients is therefore
$O(x^{-1}\log^{s+v-s}x)=O(x^{-1}\log^{v}x)$. ∎

The leading coefficient of $P_{v}$ ($s=0$ above) is

$\sum_{m_{1},\ldots,m_{e}=1}^{\infty}\mu(m_{1})\cdots\mu(m_{e})\frac{1}{M_{1}\cdots
M_{v}}=\rho(G)$

that will be further discussed in Section 3 and Section 4.

Now we consider the reciprocal sum over $G$-wise coprime tuples with some
hyperbolic constraints $\prod_{j\in A_{i}}a_{j}\leq x$ for some subsets
$A_{i}\subseteq V$, $i=1,\ldots k$ such that $\cup A_{i}=V$. As noted in [AH],
the sums over $m_{i}$ can be extended to all tuples of positive integers. If
$m_{i}>x$ for some $m_{i}$, then the inner sum over $a_{j}$ with
$j\in\epsilon_{i}$ vanishes. Thus, we are considering the sum

$\displaystyle\sum_{\begin{subarray}{c}{a_{1},\ldots,a_{v}\leq x}\\\
{G\textrm{-wise coprime}}\\\ {\forall_{i\leq k},\ \prod_{j\in A_{i}}a_{j}\leq
x}\end{subarray}}$ $\displaystyle\frac{1}{a_{1}\cdots
a_{v}}=\sum_{m_{1},\ldots,m_{e}=1}^{\infty}\mu(m_{1})\cdots\mu(m_{e})\sum_{\begin{subarray}{c}{a_{1},\ldots,a_{v}\leq
x}\\\ {\forall_{r},M_{r}|a_{r}}\\\ {\forall_{i\leq k},\ \prod_{j\in
A_{i}}a_{j}\leq x}\end{subarray}}\frac{1}{a_{1}\cdots a_{v}}$
$\displaystyle=\sum_{m_{1},\ldots,m_{e}=1}^{\infty}\mu(m_{1})\cdots\mu(m_{e})\frac{1}{M_{1}\cdots
M_{v}}\sum_{\begin{subarray}{c}{\forall_{r},b_{r}\leq\frac{x}{M_{r}}}\\\
{\forall_{i\leq k},\ \prod_{j\in A_{i}}b_{j}\leq x/\prod_{j\in
A_{i}}M_{j}}\end{subarray}}\frac{1}{b_{1}\cdots b_{v}}$
$\displaystyle=\sum_{m_{1},\ldots,m_{e}=1}^{\infty}\mu(m_{1})\cdots\mu(m_{e})\frac{1}{M_{1}\cdots
M_{v}}\sum_{\forall_{i\leq k},\ \prod_{j\in A_{i}}b_{j}\leq x/\prod_{j\in
A_{i}}M_{j}}\frac{1}{b_{1}\cdots b_{v}}.$

In the last sum, the conditions $\forall_{r},b_{r}\leq x/M_{r}$ are dropped
since the hyperbolic constraints imply them. The inner sum is more difficult
than (9) due to the hyperbolic constraints. However, if we focus on the main
term, then the following result is obtained by an elementary method.

######  Theorem 2.2.

We have

$\sum_{\begin{subarray}{c}{a_{1},\ldots,a_{v}\leq x}\\\ {G\textrm{-wise
coprime}}\\\ {\forall_{i\leq k},\ \prod_{j\in A_{i}}a_{j}\leq
x}\end{subarray}}\frac{1}{a_{1}\cdots
a_{v}}=\rho(G)\mathrm{vol}(D)\log^{v}x+O(\log^{v-1}x)$

where $D$ is a convex polytope defined by the hyperbolic constraints
$\\{A_{i}\\}_{i\leq k}$,

$D=\left\\{(t_{1},\ldots,t_{v})\in[0,\infty)^{v}\ \bigg{|}\forall_{i\leq k},\
\sum_{j\in A_{i}}t_{j}\leq 1\right\\}.$

###### Proof.

We begin with

$\sum_{\begin{subarray}{c}{a_{1},\ldots,a_{v}\leq x}\\\ {G\textrm{-wise
coprime}}\\\ {\forall_{i\leq k},\ \prod_{j\in A_{i}}a_{j}\leq
x}\end{subarray}}\frac{1}{a_{1}\cdots
a_{v}}=\sum_{m_{1},\ldots,m_{e}=1}^{\infty}\frac{\mu(m_{1})\cdots\mu(m_{e})}{M_{1}\cdots
M_{v}}\sum_{\forall_{i\leq k},\ \prod_{j\in A_{i}}b_{j}\leq x/\prod_{j\in
A_{i}}M_{j}}\frac{1}{b_{1}\cdots b_{v}}.$ (11)

Recall that for any $1\leq t$, we have

$\sum_{b\leq t}\frac{1}{b}=\log
t+O\left(1\right)=\int_{1}^{t}\frac{1}{b}db+O\left(1\right).$

Using this for the rightmost variable $b_{v}$, we replace the summation by the
integral over corresponding restrictions, then interchange the integral to the
leftmost and repeat until we replace all summations by the integrals. At each
step of this repetition, the error term $O(1)$ contributes to
$O(\log^{v-1}x)$. The inner sum becomes

$\int_{\begin{subarray}{c}{\forall_{r},1\leq b_{r}}\\\ {\forall_{i\leq k},\
\prod_{j\in A_{i}}b_{j}\leq x/\prod_{j\in
A_{i}}M_{j}}\end{subarray}}\frac{1}{b_{1}\cdots b_{v}}db_{1}\cdots
db_{v}+O(\log^{v-1}x).$

Applying the change of variables $\log b_{r}=t_{r}$ and $t_{r}/\log x=u_{r}$,
the integral becomes

$\displaystyle\int_{\begin{subarray}{c}{\forall_{r},0\leq t_{r}}\\\
{\forall_{i\leq k},\sum_{j\in A_{i}}t_{j}\leq\ \log x-\sum_{j\in A_{i}}\log
M_{j}}\end{subarray}}dt_{1}\cdots dt_{v}$
$\displaystyle=(\log^{v}x)\int_{\begin{subarray}{c}{\forall_{r},0\leq
u_{r}}\\\ {\forall_{i\leq k},\sum_{j\in A_{i}}u_{j}\leq\ 1-\frac{\sum_{j\in
A_{i}}\log M_{j}}{\log x}}\end{subarray}}du_{1}\cdots du_{v}$
$\displaystyle=(\log^{v}x)\left(\mathrm{vol}(D)+O\left(\frac{\sum_{j\leq
v}\log M_{j}}{\log x}\right)\right)$

Inserting this into the sums over $m_{i}$, we obtain the result. ∎

##  3 Coprimality Graph and Proof of Theorem 1.1

For any graph $G=(V,E)$ with $|V|=v$, $|E|=e$, it is proved in [AH,
Proposition 2] that the following functions
$f_{G}:\mathbb{N}\rightarrow\mathbb{Z}$ and
$f_{G}^{+}:\mathbb{N}\rightarrow\mathbb{Z}$ are multiplicative where

$f_{G}(m)=\sum_{M_{1}\cdots M_{v}=m}\mu(m_{1})\cdots\mu(m_{e}),\ \
f_{G}^{+}(m)=\sum_{M_{1}\cdots M_{v}=m}|\mu(m_{1})\cdots\mu(m_{e})|.$

Also, [AH] shows that $f_{G}(1)=f_{G}^{+}(1)=1$, $f_{G}(p)=f_{G}^{+}(p)=0$,
$f_{G}(p^{2})=-e$, $f_{G}^{+}(p^{2})=e$, and
$f_{G}(p^{\alpha})=f_{G}^{+}(p^{\alpha})=0$, for all $\alpha>v$. Thus, we
obtain the absolutely convergent Dirichlet series

$\sum_{m=1}^{\infty}\frac{f_{G}(m)}{m^{s}}=\prod_{p\in\mathcal{P}}\left(1+\frac{f_{G}(p^{2})}{p^{2s}}+\cdots+\frac{f_{G}(p^{v})}{p^{vs}}\right),\
\ \Re(s)>\frac{1}{2}$

and

$\sum_{m=1}^{\infty}\frac{f_{G}^{+}(m)}{m^{s}}=\prod_{p\in\mathcal{P}}\left(1+\frac{f_{G}^{+}(p^{2})}{p^{2s}}+\cdots+\frac{f_{G}^{+}(p^{v})}{p^{vs}}\right),\
\ \Re(s)>\frac{1}{2}$

By [AH, Lemma 3], $f_{G}(p^{a})$ depends only on $G$ and $a$, not on $p$. Then
we have $w=w(G)>0$ such that

$\sum_{m\leq x}\frac{|f_{G}(m)|}{\sqrt{m}}\leq\sum_{m\leq
x}\frac{f_{G}^{+}(m)}{\sqrt{m}}\leq\prod_{p\leq
x}\left(1+\frac{w}{p}\right)\leq C\log^{w}x.$

Thus, the series $\sum_{m=1}^{\infty}\frac{f_{G}(m)}{m}$ is absolutely
convergent. Then the number $\rho(G)$ defined in [AH] given by the series

$\rho(G)=\sum_{m_{1},\ldots,m_{e}=1}^{\infty}\frac{\mu(m_{1})\cdots\mu(m_{e})}{M_{1}\cdots
M_{v}}=\sum_{m=1}^{\infty}\frac{f_{G}(m)}{m}$

is absolutely convergent.

For each $F\subseteq E$, let $v(F)$ be the number of non-isolated vertices of
$F$. By [AH, Lemma 3], we have

$\rho(G)=\prod_{p\in\mathcal{P}}Q_{G}\left(\frac{1}{p}\right)$

where $Q_{G}(x)$ is a polynomial defined by

$Q_{G}(x)=\sum_{F\subseteq
E}(-1)^{|F|}x^{v(F)}=1+c_{2}x^{2}+\cdots+c_{v}x^{v}.$

For each prime $p$ and $a\geq 2$, we have $c_{a}=f_{G}(p^{a})$. Moreover,
$c_{2}=-e$ by [AH, Section 2].

By [H2, Theorem 1], we have another expression for $\rho(G)$.

$\rho(G)=\prod_{p\in\mathcal{P}}\left(\sum_{m=0}^{v}i_{m}(G)\left(1-\frac{1}{p}\right)^{v-m}\frac{1}{p^{m}}\right)$

where $i_{m}(G)$ is the number of $S\subseteq V$ with $|S|=m$ such that no two
vertices are connected by an edge in $G$ (such a set is called independent
set). In view of this expression, $\rho(G)>0$ for any graph with $v\geq 1$.
Analyzing the proofs of these papers, we have the following identity

$Q_{G}(x)=\sum_{F\subseteq
E}(-1)^{|F|}x^{v(F)}=\sum_{m=0}^{v}i_{m}(G)\left(1-x\right)^{v-m}x^{m}$

for $x=1/p$ for each prime $p$. This gives a proof that the above holds for
any $x$. We may have an interesting combinatorial proof to the above identity.
However, this is not the main topic of our result. We do not give the
combinatorial proof here.

###  3.1 The leading terms of $S_{k}$ for $k=2,3$

For $k=2$, recall by putting $a_{3}=(n_{1},n_{2})$, $n_{1}=a_{1}a_{3}$, and
$n_{2}=a_{2}a_{3}$ that

$\sum_{n_{1},n_{2}\leq
x}\frac{1}{[n_{1},n_{2}]}=\sum_{\begin{subarray}{c}{a_{1},a_{2},a_{3}\leq
x}\\\ {G\textrm{-wise coprime}}\\\ {a_{1}a_{3},a_{2}a_{3}\leq
x}\end{subarray}}\frac{1}{a_{1}a_{2}a_{3}}=\sum_{e\leq
x}\mu(e)\sum_{\begin{subarray}{c}{a_{1},a_{2},a_{3}\leq x}\\\
{e|a_{1},e|a_{2}}\\\ {a_{1}a_{3},a_{2}a_{3}\leq
x}\end{subarray}}\frac{1}{a_{1}a_{2}a_{3}}$

where $G$ is the graph $G=(V,E)$ with $V=\\{1,2,3\\}$ and $E=\\{(1,2)\\}$. It
is easy to see that $\rho(G)=\frac{6}{\pi^{2}}$ and
$\textrm{vol}(D)=\frac{1}{3}$ where $D$ is $\\{(x,y,z)\in[0,1]^{3}|x+y\leq 1,\
x+z\leq 1\\}$.

We give a detailed explanation of how the inner sum (which is a triple sum in
case $k=2$) in (11) is replaced by the triple integral at a cost of an error
term. Let $A,B\geq 1$. In the following, $u\wedge v=\min(u,v)$ and we use
indices $a$, $b$, and $c$ for simplicity of notations. We have

$\displaystyle\sum_{\begin{subarray}{c}{ac\leq x/A}\\\ {bc\leq
x/B}\end{subarray}}\frac{1}{abc}$
$\displaystyle=\sum_{a\leq\frac{x}{A}}\sum_{b\leq\frac{x}{B}}\sum_{c\leq\frac{x}{Aa}\wedge\frac{x}{Bb}}\frac{1}{abc}$
$\displaystyle=\sum_{a\leq\frac{x}{A}}\sum_{b\leq\frac{x}{B}}\frac{1}{ab}\left(\int_{1}^{\frac{x}{Aa}\wedge\frac{x}{Bb}}\frac{1}{c}\
dc+O(1)\right)$
$\displaystyle=\int_{1}^{\frac{x}{A}\wedge\frac{x}{B}}\frac{1}{c}\sum_{a\leq\frac{x}{Ac}}\frac{1}{a}\sum_{b\leq\frac{x}{Bc}}\frac{1}{b}\
dc+O(\log^{2}x)$
$\displaystyle=\int_{1}^{\frac{x}{A}\wedge\frac{x}{B}}\frac{1}{c}\sum_{a\leq\frac{x}{Ac}}\frac{1}{a}\left(\int_{1}^{\frac{x}{Bc}}\frac{1}{b}\
db+O(1)\right)dc+O(\log^{2}x)$
$\displaystyle=\int_{1}^{\frac{x}{B}}\frac{1}{b}\int_{1}^{\frac{x}{A}\wedge\frac{x}{Bb}}\frac{1}{c}\sum_{a\leq\frac{x}{Ac}}\frac{1}{a}\
dcdb+O(\log^{2}x)$
$\displaystyle=\int_{1}^{\frac{x}{B}}\frac{1}{b}\int_{1}^{\frac{x}{A}\wedge\frac{x}{Bb}}\frac{1}{c}\left(\int_{1}^{\frac{x}{Ac}}\frac{1}{a}\
da+O(1)\right)dcdb+O(\log^{2}x)$
$\displaystyle=\int_{1}^{\frac{x}{B}}\int_{1}^{\frac{x}{A}\wedge\frac{x}{Bb}}\int_{1}^{\frac{x}{Ac}}\frac{1}{bca}\
dadcdb+O(\log^{2}x)$ $\displaystyle=\iiint\limits_{\begin{subarray}{c}{ac\leq
x/A}\\\ {bc\leq x/B}\end{subarray}}\frac{1}{abc}dadbdc+O(\log^{2}x).$

Thus, by Theorem 2.2, we have

$S_{2}(x)=\sum_{n_{1},n_{2}\leq
x}\frac{1}{[n_{1},n_{2}]}=\frac{2}{\pi^{2}}\log^{3}x+O(\log^{2}x).$

Applying Dirichlet hyperbola method [Ti, (12.1.4)] to the inner sum over
$a_{r}$, $r\leq 3$, it is also possible to obtain a polynomial $P_{3}(x)$ of
degree $3$ such that

$S_{2}(x)=P_{3}(\log x)+O\left(\frac{\log^{2}x}{\sqrt{x}}\right).$ (12)

A proof of (12) was given in [HLT, Theorem 2.1]. The main method is Dirichlet
hyperbola method, but the approach is different from ours.

For $k=3$, we write $d=(n_{1},n_{2},n_{3})$, $n_{1}=afgd$, $n_{2}=befd$,
$n_{3}=cegd$, and $[n_{1},n_{2},n_{3}]=abcdefg$. This decomposition was used
in [FF]. Thus, $S_{3}(x)$ becomes

$\sum_{\begin{subarray}{c}{a,b,c,d,e,f,g\leq x}\\\ {G\textrm{-wise
coprime}}\\\ {afgd,befd,cegd\leq x}\end{subarray}}\frac{1}{abcdefg}$

where $G=(V,E)$ with $V=\\{a,b,c,d,e,f,g\\}$ and

$E=\\{(a,b),(a,c),(a,e),(b,c),(b,g),(c,f),(e,g),(e,f),(f,g)\\}.$

Therefore, by Theorem 2.2, we have

$S_{3}(x)=\rho(G)\textrm{vol}(D)\log^{7}x+O(\log^{6}x).$

Here, $D$ is a $7$-dimensional polytope

$\\{(a,b,c,d,e,f,g)\in[0,1]^{7}|a+f+g+d\leq 1,b+e+f+d\leq 1,c+e+g+d\leq 1\\}.$

The volume of this polytope is $11/3360$ by SageMath computation.

Figure 1: Coprimality graphs for $k=2,3$ (GeoGebra)

We compute $\rho(G)$ by Hu’s expression. We have $i_{0}(G)=1$ due to the empty
set, $i_{1}(G)=7$ due to the vertices, $i_{2}(G)=12$ due to the $6$ pairs of
vertices without $d$ which do not form edges in $G$ and $6$ pairs of vertices
with $d$, $i_{3}(G)=6$ due to adjoining $d$ to the $6$ pairs of vertices
without $d$, and $i_{m}(G)=0$ if $m\geq 4$. We obtain an Euler product and an
approximation by Python.

$\rho(G)=\prod_{p\in\mathcal{P}}\left(1-\frac{9}{p^{2}}+\frac{16}{p^{3}}-\frac{9}{p^{4}}+\frac{1}{p^{6}}\right)\approx
0.04932167.$

Therefore,

$\lim_{x\rightarrow\infty}\frac{S_{3}(x)}{\log^{7}x}=\rho(G)\textrm{vol}(D)=\frac{11}{3360}\prod_{p\in\mathcal{P}}\left(1-\frac{9}{p^{2}}+\frac{16}{p^{3}}-\frac{9}{p^{4}}+\frac{1}{p^{6}}\right)\approx
0.00016147.$

###  3.2 Coprimality graphs and the leading terms of $S_{k}(x)$, $U_{k}(x)$
and $V_{k}(x)$ for all $k\geq 2$

The coprimality graph for $k=2$ is $G_{2}=(\\{1,2,3\\},\\{(1,2)\\})$ as in
Figure 1. We construct $G_{k}$ inductively. Assume that
$G_{k}=(V_{k},E_{k})=(\\{1,\ldots,2^{k}-1\\},\\{\varepsilon_{1},\ldots,\varepsilon_{e_{k}}\\})$
is the coprimality graph for some $k\geq 2$. Regard the labels of vertices
with $k$-bit binary strings that contains at least one $1$,

$V_{k}=\\{1=0\cdots 01,2=0\cdots 10,3=0\cdots 11,\ldots,v_{k}=2^{k}-1=1\cdots
11\\}.$

Figure 2: Vertex labeling for $k=2,3$ (GeoGebra)

The hyperbolic constraints $\\{A_{k,i}\\}_{i\leq k}$ are given by the rule
that $A_{k,i}$ is the set of elements in $V_{k}$ such that $i$-th binary digit
(from the right) is $1$. Then for each $i\leq k$, let

$n_{i}=\prod_{j\in A_{k,i}}a_{j}\leq x.$

Our inductive hypothesis is that under the coprimality conditions
$E_{k}=\\{\varepsilon_{1},\ldots,\varepsilon_{e_{k}}\\}$ imposed on
$(a_{1},\ldots,a_{v_{k}})\in(\mathbb{N}\cap[1,x])^{v_{k}}$, we have

$[n_{1},\ldots,n_{k}]=\prod_{j\in V_{k}}a_{j},\ \
(n_{1},\ldots,n_{k})=a_{v_{k}},$

hence

$S_{k}(x)=\sum_{n_{1},\ldots,n_{k}\leq
x}\frac{1}{[n_{1},\ldots,n_{k}]}=\sum_{\begin{subarray}{c}{a_{1},\ldots,a_{v_{k}}\leq
x}\\\ {G_{k}\textrm{-wise coprime}}\\\ {\forall_{i\leq k},\prod_{j\in
A_{k,i}}a_{j}\leq x}\end{subarray}}\frac{1}{a_{1}\cdots a_{v_{k}}},$

and

$U_{k}(x)=\sum_{\begin{subarray}{c}{n_{1},\ldots,n_{k}\leq x}\\\
{(n_{1},\ldots,n_{k})=1}\end{subarray}}\frac{1}{[n_{1},\ldots,n_{k}]}=\sum_{\begin{subarray}{c}{a_{1},\ldots,a_{v_{k}}\leq
x,\ a_{v_{k}}=1}\\\ {G_{k}\textrm{-wise coprime}}\\\ {\forall_{i\leq
k},\prod_{j\in A_{k,i}}a_{j}\leq x}\end{subarray}}\frac{1}{a_{1}\cdots
a_{v_{k}}}.$

Let $V_{k+1}=\\{1,\ldots,v_{k+1}=2^{k+1}-1\\}$ be the set of ($k+1$)-bit
binary strings that contains at least one $1$. Then we may also obtain
$V_{k+1}$ by concatenating $1$ on the left for each element in $V_{k}$ and
including $2^{k}=10\cdots 0$. Let $A_{k+1,i}=A_{k,i}\cup\\{2^{k}+j\ |j\in
A_{k,i}\\}$ for each $i\leq k$ and let $A_{k+1,k+1}$ be the set of elements in
$V_{k+1}$ starting with $1$. This process of concatenating and including one
element can be viewed as introducing

$n_{k+1}=\prod_{j\in A_{k+1,k+1}}a_{j}\leq x.$

For each $i\leq k$, we redefine $n_{i}$’s as

$n_{i}=\prod_{j\in A_{k+1,i}}a_{j}\leq x.$

For each edge $(j,\ell)\in E_{k}$, let
$e_{j,\ell}=\\{2^{k}+j,j\\}\times\\{2^{k}+\ell,\ell\\}$. We write

$E^{\prime}_{k}=\bigcup_{(j,\ell)\in E_{k}}e_{j,\ell}.$

The edges in $E^{\prime}_{k}$ extend the existing edges. By the inductive
hypothesis,

$[n_{1},\ldots,n_{k}]=\prod_{j\leq 2^{k+1}-1,j\neq 2^{k}}a_{j}.$

For each $i\leq k$, let $P(n_{i},n_{k+1})$ be a product of $a_{j}$ where $j$’s
binary digits start with $1$ and $i$-th digit (from the right) is $1$. That
is,

$P(n_{i},n_{k+1})=\prod_{j\in A_{k+1,k+1}\cap A_{k+1,i}}a_{j}.$

We want to have $P(n_{i},n_{k+1})=(n_{i},n_{k+1})$. To this end, we need to
impose new coprimality conditions from

$\left(\frac{n_{i}}{P(n_{i},n_{k+1})},\frac{n_{k+1}}{P(n_{i},n_{k+1})}\right)=1,\
\forall i\leq k.$

Thus, the newly imposed coprimality conditions for each $i\leq k$ are between
the following.

$\mathbf{1.}$ $a_{j}$’s with $j$’s binary digits starting with $0$ and the
$i$-th digit is $1$.

$\mathbf{2.}$ $a_{j}$’s with $j$’s binary digits starting with $1$ and the
$i$-th digit is $0$.

Let $E^{\prime}_{k+1}$ be the set of edges from these new conditions
$(j_{1},j_{2})\in V_{k+1}^{2}$ where $j_{1}$ from 1 and $j_{2}$ from 2 so that

$E^{\prime}_{k+1}=\bigcup_{i\leq k}\left(A_{k+1,k+1}^{c}\cap
A_{k+1,i}\right)\times\left(A_{k+1,k+1}\cap A_{k+1,i}^{c}\right).$

Then we include these into the set of edges,

$E_{k+1}=E^{\prime}_{k}\cup E^{\prime}_{k+1}.$

By De Morgan’s law,

$([n_{1},\ldots,n_{k}],n_{k+1})=[(n_{1},n_{k+1}),\ldots,(n_{k},n_{k+1})].$

By the edges in $E^{\prime}_{k+1}$, we have for each $i\leq k$,

$(n_{i},n_{k+1})=P(n_{i},n_{k+1}).$

By the edges in $E^{\prime}_{k}$ and the inductive hypothesis, we have

$([n_{1},\ldots,n_{k}],n_{k+1})=[P(n_{1},n_{k+1}),\ldots,P(n_{k},n_{k+1})]=\prod_{2^{k}+1\leq
j\leq 2^{k+1}-1}a_{j}.$

Hence, we obtain

$[n_{1},\ldots,n_{k+1}]=\left[[n_{1},\ldots,n_{k}],n_{k+1}\right]=\frac{[n_{1},\ldots,n_{k}]n_{k+1}}{\left([n_{1},\ldots,n_{k}],n_{k+1}\right)}=\prod_{j\in
V_{k+1}}a_{j}.$

Moreover, we obtain by the edges in $E^{\prime}_{k+1}$,

$(n_{1},\ldots,n_{k+1})=\left((n_{1},\ldots,n_{k}),n_{k+1}\right)=a_{2^{k+1}-1}.$

We have constructed the coprimality graph $G_{k+1}$ so that

$S_{k+1}(x)=\sum_{n_{1},\ldots,n_{k+1}\leq
x}\frac{1}{[n_{1},\ldots,n_{k+1}]}=\sum_{\begin{subarray}{c}{a_{1},\ldots,a_{v_{k+1}}\leq
x}\\\ {G_{k+1}\textrm{-wise coprime}}\\\ {\forall_{i\leq k+1},\prod_{j\in
A_{k+1,i}}a_{j}\leq x}\end{subarray}}\frac{1}{a_{1}\cdots a_{v_{k+1}}}$

and

$U_{k+1}(x)=\sum_{\begin{subarray}{c}{n_{1},\ldots,n_{k+1}\leq x}\\\
{(n_{1},\ldots,n_{k+1})=1}\end{subarray}}\frac{1}{[n_{1},\ldots,n_{k+1}]}=\sum_{\begin{subarray}{c}{a_{1},\ldots,a_{v_{k+1}}\leq
x,\ a_{v_{k+1}}=1}\\\ {G_{k+1}\textrm{-wise coprime}}\\\ {\forall_{i\leq
k+1},\prod_{j\in A_{k+1,i}}a_{j}\leq x}\end{subarray}}\frac{1}{a_{1}\cdots
a_{v_{k+1}}}.$

This completes the construction of $n_{i}$, $i\leq k$ through $a_{j}$’s so
that the coprimality conditions are satisfied. Let $j_{i}$ be the $i$-th
binary digit from the right. We may also obtain $a_{j}$’s directly in terms of
$n_{i}$, $i\leq k$ by setting

$a_{j}=\prod_{p\in\mathcal{P}}p^{\nu_{p}(a_{j})}$

where $\nu_{p}(a_{j})$ is the Euclidean length of the interval

$\bigcap_{\begin{subarray}{c}{i\leq k}\\\
{j_{i}=1}\end{subarray}}[0,\nu_{p}(n_{i})]\bigcap_{\begin{subarray}{c}{i\leq
k}\\\ {j_{i}=0}\end{subarray}}[0,\nu_{p}(n_{i})]^{c}.$

Thus,

$\nu_{p}(a_{j})=\min_{\begin{subarray}{c}{i\leq k}\\\
{j_{i}=1}\end{subarray}}\nu_{p}(n_{i})-\max_{\begin{subarray}{c}{i\leq k}\\\
{j_{i}=0}\end{subarray}}\nu_{p}(n_{i}).$

Then the coprimality conditions listed in $E_{k}$ are satisfied.

Hence, the leading term of (6) follows from Theorem 2.2 with
$c_{k}=\rho(G_{k})\mathrm{vol}(D_{k})$ where

$D_{k}=\left\\{(u_{1},\ldots
u_{v_{k}})\in[0,\infty)^{v_{k}}\bigg{|}\forall_{i\leq k},\sum_{j\in
A_{k,i}}u_{j}\leq 1\right\\}.$

That is,

$S_{k}(x)=\rho(G_{k})\mathrm{vol}(D_{k})\log^{2^{k}-1}x+O(\log^{2^{k}-2}x).$

To find the leading term of (7), let $V_{k}-\\{v_{k}\\}=V_{k}^{*}$ and
$G_{k}^{*}=(V_{k}^{*},E_{k})$. Our construction guarantees that $E_{k}$ does
not contain any edges connecting to the vertex $v_{k}$. The set of edges of
$G_{k}^{*}$ and that of $G_{k}$ are identical. Then we have by [AH],

$\rho(G_{k})=\rho(G_{k}^{*}).$

For $U_{k}(x)$, we use

$U_{k}(x)=\sum_{\begin{subarray}{c}{n_{1},\ldots,n_{k}\leq x}\\\
{(n_{1},\ldots,n_{k})=1}\end{subarray}}\frac{1}{[n_{1},\ldots,n_{k}]}=\sum_{\begin{subarray}{c}{a_{1},\ldots,a_{v_{k}}\leq
x,\ a_{v_{k}}=1}\\\ {G_{k}\textrm{-wise coprime}}\\\ {\forall_{i\leq
k},\prod_{j\in A_{k,i}}a_{j}\leq x}\end{subarray}}\frac{1}{a_{1}\cdots
a_{v_{k}}}.$

The sum on the right side equals

$\sum_{\begin{subarray}{c}{a_{1},\ldots,a_{v_{k}-1}\leq x}\\\
{G_{k}^{*}\textrm{-wise coprime}}\\\ {\forall_{i\leq k},\prod_{j\in
A_{k,i}^{*}}a_{j}\leq x}\end{subarray}}\frac{1}{a_{1}\cdots a_{v_{k}-1}}$

where $A_{k,i}^{*}=A_{k,i}-\\{v_{k}\\}$ for each $i\leq k$. By Theorem 2.2,
the sum equals

$U_{k}(x)=\rho(G_{k}^{*})\textrm{vol}(D_{k}^{*})\log^{2^{k}-2}x+O(\log^{2^{k}-3}x)$

where

$D_{k}^{*}=\left\\{(u_{1},\ldots
u_{v_{k}-1})\in[0,\infty)^{v_{k}-1}\bigg{|}\forall_{i\leq k},\sum_{j\in
A_{k,i}^{*}}u_{j}\leq 1\right\\}.$

Interchanging the volume integral for $D_{k}$ and put the integral over
$u_{v_{k}}$ the leftmost, we have

$D_{k}=\int_{0}^{1}(1-u_{v_{k}})^{v_{k}-1}\textrm{vol}(D_{k}^{*})\
du_{v_{k}}=\frac{1}{v_{k}}\textrm{vol}(D_{k}^{*}).$

Hence, the leading term of (7) follows.

Clearly for any $k\geq 2$, we have $\rho(G_{k})>\rho(G_{k+1})$ by observing
that

$\rho(G_{k})=\rho((V_{k+1},E_{k}))>\rho((V_{k+1},E_{k+1}))=\rho(G_{k+1}).$

Consider a standard embedding of $D_{k}$ into $[0,\infty)^{v_{k+1}}$. We see
that

$\left(\forall_{i\leq k+1},\sum_{j\in A_{k+1,i}}u_{j}\leq 1\right)\textrm{
implies }\left(\forall_{i\leq k},\sum_{j\in A_{k,i}}u_{j}\leq 1\textrm{ and
}\sum_{j\in A_{k+1,k+1}}u_{j}\leq 1\right).$

Thus, for any $k\geq 2$, we have
$\textrm{vol}(D_{k+1})\leq\frac{1}{(2^{k})!}\textrm{vol}(D_{k})$, yielding
$c_{k}>c_{k+1}$. This implies

$\mathrm{vol}(D_{k})\leq\frac{1}{(2^{k-1})!(2^{k-2})!\cdots(2^{2})!\cdot 3}.$

Therefore, $c_{k}\rightarrow 0$ quite rapidly as $k\rightarrow\infty$.

To obtain the leading term of $V_{k}(x)$, we begin with isolating vertices
labeled as powers of $2$. We write $j\neq 2^{t}$ to indicate that $j$ is not a
power of $2$. Then we have

$\displaystyle V_{k}(x)$ $\displaystyle=\sum_{n_{1},\ldots,n_{k}\leq
x}\frac{n_{1}\cdots n_{k}}{[n_{1},\ldots,n_{k}]}$
$\displaystyle=\sum_{\begin{subarray}{c}{a_{1},\ldots,a_{v_{k}}\leq x}\\\
{G_{k}\textrm{-wise coprime}}\\\ {\forall_{i\leq k},\prod_{j\in
A_{k,i}}a_{j}\leq x}\end{subarray}}\frac{\prod\limits_{i\leq
k}\prod\limits_{j\in A_{k,i}}a_{j}}{a_{1}\ldots
a_{v_{k}}}=\sum_{\begin{subarray}{c}{a_{1},\ldots,a_{v_{k}}\leq x}\\\
{G_{k}\textrm{-wise coprime}}\\\ {\forall_{i\leq k},\prod\limits_{j\in
A_{k,i}}a_{j}\leq x}\end{subarray}}\frac{\prod\limits_{i\leq
k}\prod\limits_{j\in
A_{k,i}-\\{2^{i-1}\\}}a_{j}}{\prod\limits_{\begin{subarray}{c}{j\in V_{k}}\\\
{j\neq 2^{t}}\end{subarray}}a_{j}}$
$\displaystyle=\sum_{\begin{subarray}{c}{\forall_{j\leq v_{k},j\neq
2^{t}},a_{j}\leq x}\\\ {G_{k}\textrm{-wise coprime}}\\\ {\forall_{i\leq
k},\prod\limits_{j\in A_{k,i}-\\{2^{i-1}\\}}a_{j}\leq
x}\end{subarray}}\frac{\prod\limits_{i\leq k}\prod\limits_{j\in
A_{k,i}-\\{2^{i-1}\\}}a_{j}}{\prod\limits_{\begin{subarray}{c}{j\in V_{k}}\\\
{j\neq 2^{t}}\end{subarray}}a_{j}}\prod_{i\leq k}\sum_{a_{2^{i-1}}\leq
x/\prod\limits_{j\in A_{k,i}-\\{2^{i-1}\\}}a_{j}}1$
$\displaystyle=\sum_{m_{1},\ldots,m_{e_{k}}=1}^{\infty}\prod\limits_{j\leq
e_{k}}\mu(m_{j})\sum_{\begin{subarray}{c}{\forall_{j\leq v_{k},j\neq
2^{t}},M_{j}b_{j}\leq x}\\\ {\forall_{i\leq k},\prod\limits_{j\in
A_{k,i}-\\{2^{i-1}\\}}(M_{j}b_{j})\leq
x}\end{subarray}}\frac{\prod\limits_{i\leq k}\prod\limits_{j\in
A_{k,i}-\\{2^{i-1}\\}}(M_{j}b_{j})}{\prod\limits_{\begin{subarray}{c}{j\in
V_{k}}\\\ {j\neq 2^{t}}\end{subarray}}(M_{j}b_{j})}\prod_{i\leq
k}\sum_{b_{2^{i-1}}\leq\frac{x}{M_{2^{i-1}}\prod\limits_{j\in
A_{k,i}-\\{2^{i-1}\\}}(M_{j}b_{j})}}1$

By writing

$\sum_{b_{2^{i-1}}\leq\frac{x}{M_{2^{i-1}}\prod\limits_{j\in
A_{k,i}-\\{2^{i-1}\\}}(M_{j}b_{j})}}1=\frac{x}{M_{2^{i-1}}\prod\limits_{j\in
A_{k,i}-\\{2^{i-1}\\}}(M_{j}b_{j})}+O\left(\left[\frac{x}{M_{2^{i-1}}\prod\limits_{j\in
A_{k,i}-\\{2^{i-1}\\}}(M_{j}b_{j})}\right]^{1-\alpha}\right)$

for $\alpha<1/2$, the $O$-terms (without loss of generality) contribute the
following before summing over $m_{i}$’s.

$\displaystyle\sum_{\forall_{i\leq k},\prod\limits_{j\in
A_{k,i}-\\{2^{i-1}\\}}(M_{j}b_{j})\leq x}$
$\displaystyle\frac{x^{k-\alpha}}{\left[M_{1}\prod\limits_{j\in
A_{k,1}-\\{1\\}}(M_{j}b_{j})\right]^{1-\alpha}\prod\limits_{\begin{subarray}{c}{j\in
V_{k}-A_{k,1}}\\\ {j\neq
2^{t}}\end{subarray}}(M_{j}b_{j})\prod\limits_{\begin{subarray}{c}{j=2^{t}}\\\
{j>1}\end{subarray}}M_{j}}$
$\displaystyle=O\left(\frac{x^{k-\alpha}\log^{2^{k-1}-k}x\sum\limits_{n\leq
x}n^{\alpha-1}\tau_{2^{k-1}-1}(n)}{\left[M_{1}\prod\limits_{j\in
A_{k,1}-\\{1\\}}M_{j}\right]^{1-\alpha}\prod\limits_{\begin{subarray}{c}{j\in
V_{k}-A_{k,1}}\\\ {j\neq
2^{t}}\end{subarray}}M_{j}\prod\limits_{\begin{subarray}{c}{j=2^{t}}\\\
{j>1}\end{subarray}}M_{j}}\right)$
$\displaystyle=O\left(\frac{x^{k}\log^{2^{k}-k-2}x}{\left[\prod\limits_{j\in
A_{k,1}}M_{j}\right]^{1-\alpha}\prod\limits_{j\in
V_{k}-A_{k,1}}M_{j}}\right).$

We have the convergence of the sum over $m_{i}$’s due to $\alpha<1/2$. Then we
have $V_{k}(x)$ up to an error of $O(x^{k}\log^{2^{k}-k-2}x)$:

$\displaystyle V_{k}(x)$
$\displaystyle=x^{k}\sum_{m_{1},\ldots,m_{e_{k}}=1}^{\infty}\frac{\prod\limits_{j\leq
e_{k}}\mu(m_{j})}{\prod\limits_{j\leq v_{k}}M_{j}}\sum_{\forall_{i\leq
k},\prod\limits_{j\in A_{k,i}-\\{2^{i-1}\\}}(M_{j}b_{j})\leq
x}\frac{1}{\prod\limits_{\begin{subarray}{c}{j\in V_{k}}\\\ {j\neq
2^{t}}\end{subarray}}b_{j}}+O(x^{k}\log^{2^{k}-k-2}x)$
$\displaystyle=x^{k}\rho(G_{k})\mathrm{vol}(D^{**}_{k})\log^{2^{k}-k-1}x+O(x^{k}\log^{2^{k}-k-2}x)$

where $D^{**}_{k}$ is a convex polytope defined by the hyperbolic constraints
$\\{A_{i}-\\{2^{i-1}\\}\\}_{i\leq k}$,

$D^{**}_{k}=\left\\{(t_{i})_{\begin{subarray}{c}{i\leq{v_{k}}}\\\ {i\neq
2^{t}}\end{subarray}}\in[0,\infty)^{2^{k}-k-1}\ \bigg{|}\forall_{i\leq k},\
\sum_{j\in A_{k,i}-\\{2^{i-1}\\}}t_{j}\leq 1\right\\}.$

###  3.3 Multivariable Perron’s formula

A multivariable Perron’s formula relates the sum of multivariable arithmetic
function

$\sum_{n_{1},\ldots,n_{k}\leq x}f(n_{1},\ldots,n_{k})$

to an integral over some vertical lines of complex planes. A general result
yielding an asymptotic formula with power-saving error term was proved by de
la Bretèche [dlB, Theorem 1]. An effective version of Perron’s formula was
proved by Balazard, Naimi, Pétermann [BNP, Proposition 5] in which they
provided an asymptotic formula for the sum

$\sum_{n_{1},\ldots,n_{k}\leq
x}\frac{\mu(n_{1})\cdots\mu(n_{k})}{[n_{1},\ldots,n_{k}]}.$

Tóth and Zhai [TZ] proved an asymptotic formula for the average number of
subgroups of $\mathbb{Z}/m\mathbb{Z}\times\mathbb{Z}/n\mathbb{Z}$, $m,n\leq x$
by using the effective Perron’s formula. We state the effective version of
Perron’s formula [BNP, Proposition 5] here.

######  Proposition 3.1.

Let $f:\mathbb{N}^{k}\rightarrow\mathbb{C}$ be an arithmetic function of $k$
variables and $(\sigma_{a_{1}},\ldots,\sigma_{a_{k}})$ an $k$-tuple of the
abscissas of absolute convergence of the associated Dirichlet series

$F(s_{1},\ldots,s_{k})=\sum_{n_{1},\ldots,n_{k}=1}^{\infty}\frac{f(n_{1},\ldots,n_{k})}{{n_{1}}^{s_{1}}\cdots{n_{k}}^{s_{k}}}.$

We have for $x_{1}\geq 1$, $\ldots\ $, $x_{k}\geq 1$,
$\kappa_{1}>\max(0,\sigma_{a_{1}})$, $\ldots\ $,
$\kappa_{k}>\max(0,\sigma_{a_{k}})$, $T_{1}\geq 1$, $\ldots\ $, $T_{k}\geq 1$,

$\left|\sideset{}{{}^{\prime}}{\sum}_{n_{1}\leq x_{1},\ldots,n_{k}\leq
x_{k}}f(n_{1},\ldots,n_{k})-\frac{1}{(2\pi
i)^{k}}\int\displaylimits_{\kappa_{k}-iT_{k}}^{\kappa_{k}+iT_{k}}\cdots\int\displaylimits_{\kappa_{1}-iT_{1}}^{\kappa_{1}+iT_{1}}F(s_{1},\ldots,s_{k})\frac{{x_{1}}^{s_{1}}\cdots{x_{k}}^{s_{k}}}{s_{1}\cdots
s_{k}}ds_{1}\cdots ds_{k}\right|$ $\displaystyle\leq x_{1}^{\kappa_{1}}\cdots
x_{k}^{\kappa_{k}}$
$\displaystyle\sum_{n_{1},\ldots,n_{k}=1}^{\infty}\frac{|f(n_{1},\ldots,n_{k})|}{{n_{1}}^{\kappa_{1}}\cdots{n_{k}}^{\kappa_{k}}}Q_{k}\left(\frac{1}{(\pi
T_{1}|\log(x_{1}/n_{1})|)^{*}},\ldots,\frac{1}{(\pi
T_{k}|\log(x_{k}/n_{k})|)^{*}}\right)$

where the primed sum takes the value of $f(n_{1},\ldots,n_{k})/2^{m}$ if the
number of $i\leq k$ with $n_{i}=x_{i}$ is m, the multivariable polynomial
$Q_{k}$ is defined by

$Q_{k}(X_{1},\ldots,X_{k})=(1+X_{1})\cdots(1+X_{k})-1,$

and $y^{*}=\max(1,y)$.

We are interested in the case where $f$ takes the following form

$f(n_{1},\ldots,n_{k})=\sum_{\begin{subarray}{c}{\forall_{r},a_{r}\in\mathbb{N}}\\\
{G\textrm{-wise coprime}}\\\ {\forall_{i\leq k},\ \prod_{j\in
A_{i}}a_{j}=n_{i}}\end{subarray}}\frac{1}{a_{1}\cdots a_{v}}$ (13)

where $G=(V,E)=(\\{1,\ldots,v\\},\\{\epsilon_{1},\ldots,\epsilon_{e}\\})$ is a
graph and $A_{i},i\leq k$ are nonempty with $\cup_{i\leq
k}A_{i}=\\{1,\ldots,v\\}$. The associated Dirichlet series of the arithmetic
function $f$ above satisfies the following.

######  Lemma 3.1.

Let $f$ be the arithmetic function defined in (13), ${\bf
s}=(s_{1},\ldots,s_{k})\in\mathbb{C}^{k}$, and ${\bf e}_{i},i\leq k$ be the
standard basis of $\mathbb{C}^{k}$. Then

$F(s_{1},\ldots,s_{k})=\sum_{n_{1},\ldots,n_{k}=1}^{\infty}\frac{f(n_{1},\ldots,n_{k})}{{n_{1}}^{s_{1}}\cdots{n_{k}}^{s_{k}}}=H(s_{1},\ldots,s_{k})\prod_{r\leq
v}\zeta(1+\ell_{r}({\bf s}))$ (14)

converges absolutely if $\Re(s_{i})>0$ for each $i\leq k$, the Dirichlet
series of $H$ converges absolutely if $\Re(s_{i})>-\frac{1}{2k}$ for each
$i\leq k$, and the linear forms $\ell_{r}({\bf s})$ is defined by

$\ell_{r}({\bf s})=\sum_{i\leq k}\ell_{r}({\bf e}_{i})s_{i}$

where

$\ell_{r}({\bf e}_{i})=\begin{cases}1&\mbox{ if }r\in A_{i}\\\ 0&\mbox{
otherwise.}\end{cases}$

###### Proof.

The absolute convergence of $F$ when $\Re(s_{i})>0$ for each $i\leq k$ is
clear. We have by the method of Section 2,

$\displaystyle\sum_{n_{1},\ldots,n_{k}=1}^{\infty}$
$\displaystyle\frac{f(n_{1},\ldots,n_{k})}{{n_{1}}^{s_{1}}\cdots{n_{k}}^{s_{k}}}=\sum_{\begin{subarray}{c}{\forall_{r},a_{r}\in\mathbb{N}}\\\
{G\textrm{-wise coprime}}\end{subarray}}\frac{1}{a_{1}^{1+\ell_{1}({\bf
s})}\cdots a_{v}^{1+\ell_{v}({\bf s})}}$
$\displaystyle=\sum_{m_{1},\ldots,m_{e}=1}^{\infty}\frac{\mu(m_{1})\cdots\mu(m_{e})}{M_{1}^{1+\ell_{1}({\bf
s})}\cdots M_{v}^{1+\ell_{v}({\bf
s})}}\sum_{b_{1},\ldots,b_{v}=1}^{\infty}\frac{1}{b_{1}^{1+\ell_{1}({\bf
s})}\cdots b_{v}^{1+\ell_{v}({\bf s})}}.$

The inner sum over $b_{i}$ is $\prod_{r\leq v}\zeta(1+\ell_{r}({\bf s}))$. By
the analysis of the multiplicative function $f_{G}(m)$ in the beginning of
Section 3, the outer sum over $m_{i}$ defines a Dirichlet series which
converges absolutely if $\Re(1+\ell_{r}({\bf s}))>1/2$ for each $r\leq v$.
This is satisfied if $\Re(s_{i})>-\frac{1}{2k}$ for each $i\leq k$. Denote
this outer sum by $H(s_{1},\ldots,s_{k})$, then we have the result. ∎

The following are the well-known upper bound and mean value results for the
Riemann zeta function. Similar results were also used in [TZ].

######  Lemma 3.2.

Fix $\delta>0$, $M,N\geq 0$, and let $s=\sigma+it$. For any $\epsilon>0$,
there is an absolute constant $A>0$ depending on $\delta$, $\epsilon$, $M$,
and $N$ such that for all $j\leq N$, $1/2+\delta\leq\sigma\leq 1+\delta$,
$t\geq 1$, and $T\geq 1$,

$|\zeta^{(j)}(s)|\leq A|t|^{\frac{1}{3}\max(1-\sigma,0)+\epsilon}$ (15)

and

$\int_{-T}^{T}|\zeta^{(j)}(\sigma+it)|^{M}dt\leq A\
T^{\frac{\max(M-4,0)}{3}\max(1-\sigma,0)+1+\epsilon}.$ (16)

###### Proof.

If $j=0$, (15) follows from [Ti, (5.12)] and Phragmén-Lindelöf principle. For
$j>0$, the result follows by Cauchy’s integral formula. If $j=0$, use

$|\zeta(\sigma+it)|^{M}=|\zeta(\sigma+it)|^{\max(M-4,0)}|\zeta(\sigma+it)|^{\min(M,4)}.$

Then (16) follows from [Ti, Theorem 7.5] and the analogous result for the mean
values in [Ti, page 149] similar to Phragmén-Lindelöf principle. Again, the
result for $j>0$ follows by Cauchy’s integral formula. ∎

For the values of $\zeta^{(j)}(s)$ when $s$ is close to the line $\Re(s)=1$,
we have the following result.

######  Lemma 3.3.

Let $A$ be any positive constant. Fix $N\geq 0$, and let $s=\sigma+it$ with
$t\geq t_{0}$. Then there are absolute constants $B>0$ and $M>0$ depending on
$A$ and $N$ such that for all $j\leq N$,

$|\zeta^{(j)}(s)|\leq B\log^{M}t$

uniformly in the region

$1-\frac{A}{\log t}\leq\sigma\leq 2,\ t\geq t_{0}.$

###### Proof.

For $j=0$, this is [Ti, Theorem 3.5]. For $j=1$, the proof is in [Ti,
(3.5.5)]. The proofs for the higher derivatives are analogous. ∎

We establish a version of Perron’s formula for the Dirichlet series (14). The
following is a special case of [dlB, Theorem 1]. The purpose of the following
proposition is to find an explicit power-saving error term with the exponent
of $x$ is depending on $k$ and $v$. Our assumptions are more restrictive than
that of [dlB, Theorem 1]. We use the notations ${\bf
n}=(n_{1},\ldots,n_{k})\in\mathbb{N}^{k}$,
$\boldsymbol{\alpha}=(\alpha_{1},\ldots,\alpha_{k})\in[0,\infty)^{k}$, ${\bf
0}=(0,\ldots,0)\in\mathbb{C}^{k}$, ${\bf 1}=(1,\ldots,1)\in\mathbb{C}^{k}$,
${\bf
n}^{\boldsymbol{\alpha}}={n_{1}}^{\alpha_{1}}\cdots{n_{k}}^{\alpha_{k}}$, and
$J(\boldsymbol{\alpha})=\\{j\leq k\ |\ \alpha_{j}=0\\}$.

######  Proposition 3.2.

Let $f:\mathbb{N}^{k}\rightarrow\mathbb{R}$ be an arithmetic function
satisfying the following conditions.

i) The associated Dirichlet series

$F(s_{1},\ldots,s_{k})=\sum_{n_{1},\ldots,n_{k}=1}^{\infty}\frac{f(n_{1},\ldots,n_{k})}{{n_{1}}^{s_{1}}\cdots{n_{k}}^{s_{k}}}=H(s_{1},\ldots,s_{k})\prod_{r\leq
v}\zeta(1+\ell_{r}({\bf s})),$

converges absolutely if $\Re(s_{i})>0$ for each $i\leq k$, the Dirichlet
series of $H$ converges absolutely if $\Re(s_{i})>-\frac{1}{2k}$ for each
$i\leq k$, and $\ell_{r}({\bf s})$, $r\leq v$ are nonzero linear forms.

ii) There is a constant $M=M_{f}\in\mathbb{N}$ such that for all $i\leq k$ and
$0<\kappa_{j}<1$,

$\sum_{\begin{subarray}{c}{\forall_{j\leq k},n_{j}\in\mathbb{N}}\\\ {j\neq
i}\end{subarray}}\frac{|f(n_{1},\ldots,n_{i-1},n_{i},n_{i+1},\ldots,n_{k})|}{{n_{1}}^{\kappa_{1}}\cdots{n_{i-1}}^{\kappa_{i-1}}{n_{i+1}}^{\kappa_{i+1}}\cdots{n_{k}}^{\kappa_{k}}}\leq\frac{M\tau_{v}(n_{i})}{(\kappa_{1}\cdots\kappa_{i-1})^{M}\
n_{i}\ (\kappa_{i+1}\cdots\kappa_{k})^{M}}.$ (17)

where $\tau_{v}(m)=\sum_{a_{1}\cdots a_{v}=m}1$ is the divisor function.

Then we have

$\sum_{n_{1},\ldots,n_{k}\leq x}{\bf
n}^{\boldsymbol{\alpha}}f(n_{1},\ldots,n_{k})=x^{\boldsymbol{\alpha}\cdot{\bf
1}}P_{v+w-\ell}(\log x)+O(x^{\boldsymbol{\alpha}\cdot{\bf
1}-\frac{v_{*}}{C}+\epsilon})$

where $w=|J(\boldsymbol{\alpha})|$, $\ell=\mathrm{rank}\left(\\{\ell_{r}\ |\
r\leq v\\}\cup\\{{\bf e}_{j}^{*}\ |\ j\in J(\boldsymbol{\alpha}),\ j\leq
k\\}\right)$, $v_{*}=\frac{3}{6k+\max(v-4,0)}$,

Here, $C\geq 1$ will be defined in the proof.

###### Proof.

We first treat the case $\boldsymbol{\alpha}={\bf 0}\in[0,\infty)^{k}$. We
apply Proposition 3.1 with $2\leq x_{1}=\cdots=x_{k}=x\in\mathbb{N}$, $2\leq
T_{1}=\cdots=T_{k}=T\leq x$, and $0<\kappa_{j}<\frac{1}{\log x}$ for each
$j\leq k$. We treat the error term in Proposition 3.1. Without loss of
generality, consider a monomial term $X_{1}\cdots X_{r}$ of $Q_{k}$. By
$0<1/y^{*}\leq 1$ and (17), the contribution of the terms such that some
$n_{i}$, $i\leq r$ do not belong to $(x/2,2x)$ is

$\ll
x^{\kappa_{1}+\cdots+\kappa_{k}}\sum_{n_{1},\ldots,n_{k}=1}^{\infty}\frac{|f(n_{1},\ldots,n_{k})|}{{n_{1}}^{\kappa_{1}}\cdots{n_{k}}^{\kappa_{k}}T}\ll\frac{\mathcal{L}^{M}}{T}.$

For the remaining terms, without loss of generality, consider the sum over
$n_{1}\in(x/2,2x)-\\{x\\}$, $n_{2},\ldots,n_{k}\in\mathbb{N}$. By
$|\log(x/n)|\asymp|x-n|/x$ for $n\asymp x$ and (17), the sum of such terms
satisfy

$\displaystyle\ll
x^{\kappa_{1}+\cdots+\kappa_{k}}\sum_{\begin{subarray}{c}{x/2<n_{1}<2x}\\\
{n_{1}\neq
x}\end{subarray}}\sum_{n_{2},\ldots,n_{k}=1}^{\infty}\frac{|f(n_{1},\ldots,n_{k})|}{{n_{1}}^{\kappa_{1}}\cdots{n_{k}}^{\kappa_{k}}}\frac{x}{T|x-n_{1}|}$
$\displaystyle\ll\sum_{\begin{subarray}{c}{x/2<n_{1}<2x}\\\ {n_{1}\neq
x}\end{subarray}}\frac{\tau_{v}(n_{1})x\mathcal{L}^{M}}{Tn_{1}|x-n_{1}|}\ll\sum_{\begin{subarray}{c}{x/2<n_{1}<2x}\\\
{n_{1}\neq
x}\end{subarray}}\frac{\tau_{v}(n_{1})\mathcal{L}^{M}}{T|x-n_{1}|}\ll\frac{x^{\epsilon}}{T}.$

With some more work (see [BNP, Lemma 1, 2, and 3]), this error from the
remaining terms can be improved to $O(\mathcal{L}^{M}/T)$ provided that $T\leq
x^{v_{*}-\epsilon}$.

The term with $n_{1}=x$, is by $0<1/y^{*}\leq 1$ and (17),

$\displaystyle\ll x^{\kappa_{1}+\cdots+\kappa_{k}}x^{-1+\epsilon}\ll
x^{-1+\epsilon}.$

Therefore, the combined error term becomes $\ll T^{-1}\mathcal{L}^{M}$ when
$2\leq T\leq x^{v_{*}-\epsilon}$.

Also, by the assumption (17) on $f$,

$\sideset{}{{}^{\prime}}{\sum}_{n_{1},\ldots,n_{k}\leq
x}f(n_{1},\ldots,n_{k})=\sum_{n_{1},\ldots,n_{k}\leq
x}f(n_{1},\ldots,n_{k})+O(x^{-1+\epsilon}).$

By Proposition 3.1, we have

$\sum_{n_{1},\ldots,n_{k}\leq x}f(n_{1},\ldots,n_{k})=\frac{1}{(2\pi
i)^{k}}\int\displaylimits_{\kappa_{k}-iT_{k}}^{\kappa_{k}+iT_{k}}\cdots\int\displaylimits_{\kappa_{1}-iT_{1}}^{\kappa_{1}+iT_{1}}F(s_{1},\ldots,s_{k})\frac{x^{s_{1}+\cdots+s_{k}}}{s_{1}\cdots
s_{k}}ds_{1}\cdots ds_{k}+O(\frac{\mathcal{L}^{M}}{T}).$

As in [dlB, (2.11)], we choose $\kappa_{j}=\kappa^{j}/\log x$ where $\kappa$
is a positive transcendental over the smallest field $K$ containing
$\mathbb{Q}$ and all coefficients of $\ell_{r}({\bf s})$. This is to ensure
any distinct linear forms that appear in evaluating the above integral have
distinct values.

For the general $\boldsymbol{\alpha}$, we obtain by applying Proposition 3.1
to the function

$F(s_{1}-\alpha_{1},\ldots,s_{k}-\alpha_{k})$

and change of variables,

$\sum_{n_{1},\ldots,n_{k}\leq x}{\bf n}^{\boldsymbol{\alpha}}f({\bf
n})=\frac{x^{\boldsymbol{\alpha}\cdot{\bf 1}}}{(2\pi
i)^{k}}\int\displaylimits_{\kappa_{k}-iT_{k}}^{\kappa_{k}+iT_{k}}\cdots\int\displaylimits_{\kappa_{1}-iT_{1}}^{\kappa_{1}+iT_{1}}\frac{F(s_{1},\ldots,s_{k})x^{s_{1}+\cdots+s_{k}}}{(s_{1}+\alpha_{1})\cdots(s_{k}+\alpha_{k})}ds_{1}\cdots
ds_{k}+O(\frac{x^{\boldsymbol{\alpha}\cdot{\bf 1}}\mathcal{L}^{M}}{T})$

with the same choice of $\kappa_{j}$.

The steps of evaluating this integral are written in [dlB] and [BNP]. Here, we
include the outline of the method. The idea is to evaluate the integral
starting from the innermost by moving the line of integration to
$[-\delta_{1}-iT_{1},-\delta_{1}+iT_{1}]$ and using the residue theorem. We
choose $\delta_{1}$ and subsequent $\delta_{j}$’s so that the zeta functions
are evaluated close to the critical line and the mean value result (Lemma 3.2)
are applied. Note that $\zeta(1+s)$ has a simple pole at $s=0$.

$\mathbf{1.}$ Consider a $(v+w+k+1)\times k$ matrix
$L=(a_{ij})_{\begin{subarray}{c}{1\leq i\leq v+w+k+1}\\\ {1\leq j\leq
k}\end{subarray}}$ obtained by the coefficients of $\ell_{r}$, $r\leq v$ on
the first $v$ rows, and ${\bf e}_{j},\ j\in J(\boldsymbol{\alpha})$ on the
next $w$ rows. Then we put the $k\times k$ identity matrix below. The last row
is ${\bf 1}\in\mathbb{C}^{k}$. Let $L_{0}$ be the top $v+w$ rows of $L$.

$\mathbf{2.}$ Choose a row with nonzero first entry (if exists, and it is not
the last $k+1$ rows). Apply the row interchange to move this row to the first
row $R_{1}$. Apply the row interchange to move all remaining rows $R_{i}$,
$i\leq v+w$ with nonzero first entry to the rows directly below $R_{1}$. By
taking the dot product with $(0,-s_{2},\ldots,-s_{k})$ and dividing by the
nonzero first entry of these rows, we obtain the locations of poles of the
integrand with respect to the variable $s_{1}$. The poles are linear forms of
$s_{2},\ldots,s_{k}$.

$\mathbf{3.}$ We find the residues at these poles. The sum of the residues is
the value of the innermost integral up to an error caused by two horizontal
sides and the left vertical side of the rectangle with vertices
$-\delta_{1}-iT_{1}$, $\kappa_{1}-iT_{1}$, $\kappa_{1}+iT_{1}$, and
$-\delta_{1}+iT_{1}$.

$\mathbf{4.}$ We need to substitute the chosen pole in place of $s_{1}$ into
the integrand except for the function corresponding to the chosen pole. We
apply the row replacement of $R_{i}$ by $R_{i}-\frac{a_{i1}}{a_{11}}R_{1}$ to
all the rows below $R_{1}$. This will remove the leading nonzero entries of
these rows except for $R_{1}$. Then we proceed on integrating the residue with
respect to $s_{2}$.

$\mathbf{5.}$ Suppose that we are at the stage of the integral with respect to
$s_{j}$, $j\geq 2$ and the last row shows $s_{1}+\cdots+s_{k}$ after the
substitutions of $s_{1},\ldots,s_{j-1}$. The $k$ rows above the last row show
the substitutions of $s_{1},\ldots,s_{k}$ in terms of $s_{j},\ldots,s_{k}$.

Case 1: The first nonzero entry in the last row is at the position $j_{0}\geq
j$ and it is positive.

We interchange the order of integration from $ds_{j}\cdots ds_{k}$ to

$ds_{j_{0}}ds_{j}ds_{j+1}\cdots ds_{j_{0}-1}ds_{j_{0}+1}\cdots ds_{k}$

with an obvious modification when $j_{0}=j,k$. The poles with respect to
$s_{j_{0}}$ are linear forms of
$s_{j},\ldots,s_{j_{0}-1},s_{j_{0}+1},\ldots,s_{k}$. We find the residues at
these poles. Move the line of integration to
$[-\delta_{j_{0}}-iT_{j_{0}},-\delta_{j_{0}}+iT_{j_{0}}]$. The error caused by
two horizontal sides and the left vertical sides of the rectangle with
vertices
$-\delta_{j_{0}}-iT_{j_{0}},\kappa_{j_{0}}-iT_{j_{0}},\kappa_{j_{0}}+iT_{j_{0}},$
and $-\delta_{j_{0}}+iT_{j_{0}}$.

Case 2: The first nonzero entry in the last row is at the position $j_{0}\geq
j$ and it is negative.

We interchange the order of integration as in Case 1. We find the residues at
these poles. Move the line of integration to
$[\delta_{j_{0}}-iT_{j_{0}},\delta_{j_{0}}+iT_{j_{0}}]$. The error caused by
two horizontal sides and the right vertical sides of the rectangle with
vertices
$\delta_{j_{0}}-iT_{j_{0}},\kappa_{j_{0}}-iT_{j_{0}},\kappa_{j_{0}}+iT_{j_{0}},$
and $\delta_{j_{0}}+iT_{j_{0}}$.

In Case 1 and Case 2, we modify $T_{j_{0}}$ by $T_{j_{0}}+O(1)$ to avoid the
poles in the path of integration if necessary. This may result in an error of
$O(\mathcal{L}^{M}/T)$ by Lemma 3.3. We may not have all poles enclosed by the
rectangle. However, it is possible to include all poles on one side of the
vertical line $\Re s=\kappa_{j_{0}}$ at a cost of an error
$O(\mathcal{L}^{M}/T)$ due to $(s_{j_{0}}+\alpha_{j_{0}})^{-1}$ in the
integrand and by Lemma 3.3. The interchange of the order of integration causes
the reordering of the columns of $L$ to be consistent with the order of
variables. The steps 2, 3, 4, and 5 are repeated until the following Case 3
appears. If Case 3 has never appeared, then the process ends with contour
shifting and finding residues at poles about the last variable of
consideration.

Case 3: The $j_{0}$-th position of the last row is zero for all $j_{0}\geq j$.

The integrand at this stage is consisted of certain partial derivatives of
$H$, derivatives of $\zeta$, and the denominator as a product of linear
functions in $s_{j},\ldots,s_{k}$. This is due to the successive substitutions
of $s_{1},\ldots s_{j-1}$ in terms of $s_{j},\ldots,s_{k}$. Then

$s_{1}+\cdots+s_{j-1}=-s_{j}-\cdots-s_{k}$

guarantees that for each $j_{0}\geq j$, there are at least two linear
functions (one of them is $s_{j_{0}}+\alpha_{j_{0}}$) in the denominator
containing nonzero coefficients of $s_{j_{0}}$. To see this, for each
$j_{0}\geq j$, all substitutions that yields nonzero coefficients of
$s_{j_{0}}$ are from the poles of zeta functions or their derivatives. We
write $\kappa_{j}=\kappa_{j}(x)$, $\mathcal{L}^{M}=\mathcal{L}^{M}(x)$, and
$T=T(x)$ (Here, $T$ will be a fixed power of $x$). We apply the method of
[BNP, Lemma 20], let $I(x)$ be the integral of the integrand as follows:

$I(x)=\int\displaylimits_{\kappa_{k}(x)-iT_{k}(x)}^{\kappa_{k}(x)+iT_{k}(x)}\cdots\int\displaylimits_{\kappa_{j}(x)-iT_{j}(x)}^{\kappa_{j}(x)+iT_{j}(x)}(\cdots)\
ds_{j}\cdots ds_{k}$

Then for sufficiently large $x$, that is, $x\geq x_{0}$, the contour changes
(first horizontal then vertical) to produce $I(2x)$ contribute the errors of
$O(\mathcal{L}^{M}(x)/T(x))$. This gives
$|I(2x)-I(x)|=O(\mathcal{L}^{M}(x)/T(x))$. Thus, $I(2^{m}x_{0})$ forms a
Cauchy sequence and it converges to a number $I$ as $m\rightarrow\infty$.
Hence, we obtain for sufficiently large $x$ and $2^{m_{0}}x_{0}\leq
x<2^{m_{0}+1}x_{0}$,

$|I(x)-I|=O\left(\sum_{m=m_{0}}^{\infty}\frac{\mathcal{L}^{M}(2^{m}x_{0})}{T(2^{m}x_{0})}\right)=O\left(\frac{\mathcal{L}^{M}(x)}{T(x)}\right).$

$\mathbf{6.}$ The procedure stops when all variables are considered and Step 5
cannot be repeated. During this process, the matrix $L$ goes through row
replacements, row interchanges of the first $v+w$ rows, and column
interchanges.

We lay out all matrices appearing as results of these row or column
operations. There are finitely many such matrices. Let $C\geq 1$ be the
maximum of the absolute values of all entries of all matrices in this layout.
Let $0<c\leq 1$ be the minimum of the absolute values of all nonzero entries
of all matrices in this layout. We require $\kappa<c^{2}/(2C^{2}+2)$. This
choice of $\kappa$ guarantees that

$c^{2}\kappa^{j}>C^{2}(\kappa^{j+1}+\kappa^{j+2}+\cdots).$

Then for each $j_{0}\geq j$ and the vertical segment
$[\kappa_{j_{0}}-iT_{j_{0}},\kappa_{j_{0}}+iT_{j_{0}}]$ of integration, we can
decide whether a pole

$s_{j_{0}}=a_{j}s_{j}+\ldots+a_{j_{0}-1}s_{j_{0}-1}+a_{j_{0}+1}s_{j_{0}+1}+\ldots+a_{k}s_{k}$

(an obvious modification when $j_{0}=j,k$) is on the left ($j<j_{0}$ and
$a_{j}<0$) or right side ($j<j_{0}$ and $a_{j}>0$) of the segment from the
sign of $a_{j}$. If $j=j_{0}$, then the pole does not have $a_{j}s_{j}$ term
and the pole is on the left side of the segment.

As in [Ti, Theorem 12.3], we use Lemma 3.2 and find the contribution of the
contour changes to $[-\delta_{j_{0}}-iT_{j_{0}},-\delta_{j_{0}}+iT_{j_{0}}]$
or $[\delta_{j_{0}}-iT_{j_{0}},\delta_{j_{0}}+iT_{j_{0}}]$ in Step 5 Case 1
and 2. We enlarge $C$ so that $\min\limits_{\begin{subarray}{c}{1\leq j\leq
k}\\\ {\alpha_{j}>0}\end{subarray}}\alpha_{j}>\frac{1}{2kC}$, if necessary.
This is to ensure the poles are only from the zeta functions or the linear
functions on the denominator. We take $\delta_{j}=\frac{1}{2kC}+\epsilon$ for
each $j$. The errors due to the integrals over each horizontal side are

$\ll(T^{\frac{v}{6k}-1}x^{-\frac{1}{2kC}}+T^{-1})(xT)^{\epsilon}.$

The errors due to the integrals over each vertical side are

$\ll x^{-\frac{1}{2kC}}T^{\frac{1}{6k}\max(v-4,0)}(xT)^{\epsilon}.$

Noting that $\frac{v-6k}{6k}<\frac{1}{6k}\max(v-4,0)$, we take
$T=x^{\frac{v_{*}}{C}}$ to balance the error terms $T^{-1}(xT)^{\epsilon}$ and
$x^{-\frac{1}{2kC}}T^{\frac{1}{6k}\max(v-4,0)}(xT)^{\epsilon}$, then the
combined errors become

$\ll x^{\boldsymbol{\alpha}\cdot{\bf 1}-\frac{v_{*}}{C}+\epsilon}.$

During the process from 1 to 6 on these residues, a power of $\log x$ is
obtained whenever we take the derivative of the power of $x$ in the integrand.
Here, $v+w-\ell$ is the maximal time we take the derivative of powers of $x$.
∎

Remark.  If the entries of matrix $L$ and all $\alpha_{i}$’s are integers,
then we have the following explicit bounds for $C$ and $c$. Let $\|L\|$ be the
maximum of absolute values of entries of $L$. Then

$C\leq{\|L\|}^{k}k^{\frac{k}{2}}\ \textrm{ and }\
c\geq\|L\|^{-(k-1)}(k-1)^{-\frac{k-1}{2}}.$

To see this, suppose that the entry corresponding to $C$ is obtained at
$(i_{0},j_{0})$ position with $i_{0}\geq i$ and $j_{0}\geq j$. Up to this
stage, $i-1$ pivot positions are determined and $(i-1)$-th pivot position is
at $(i-1,j-1)$-th entry. Label the rows of $L_{0}$ which are selected to have
these $i-1$ pivot positions as $R_{1},\ldots,R_{i-1}$. Apply a column
interchange to move $(i_{0},j_{0})$-th entry to $(i_{0},j)$-th entry and
relabel the rows a $R_{1},\ldots,R_{i-1}$ after the column interchange. Then
apply a row interchange to move it to $(i,j)$-th entry. Label the row
containing the pivot at $(i,j)$-th entry as $R_{i}$. Insert ${\bf e}_{i}$’s
($i\leq k$) to fill up the missing pivot positions so that the pivot at
$(i,j)$-th entry is moved to $(j,j)$-th entry and all $k$ pivot positions are
determined. We consider a linear system with an augmented matrix with rows

$R_{1}+0\ {\bf e}_{k+1},\ldots,R_{i-1}+0\ {\bf e}_{k+1},R_{i}+1\ {\bf
e}_{k+1},$

and ${\bf e}_{i}$’s used for filling up the pivots. The solution to this
system has $s_{j+1}=\cdots=s_{k}=0$ and $|s_{j}|=\frac{1}{C}$ since the row
operations between $R_{1},\ldots,R_{i}$ do not change $1\ {\bf e}_{k+1}$. Let
$B$ be the coefficient matrix of this system. By Cramér’s rule

$s_{j}=\frac{|B^{\prime}|}{|B|}$

where $B^{\prime}$ is $(k-1)\times(k-1)$ minor of $B$ obtained by removing
$j$-th row and column. Since $s_{j}$ is nonzero and $L$ is integer matrix, we
have $|B^{\prime}|\geq 1$. By Hadamard’s determinant bound we have
$|B|\leq{\|L\|}^{k}k^{\frac{k}{2}}$. This yields the upper bound for $C$. The
lower bound for $c$ is obtained by the similar method with $|B|\geq 1$ and
$|B^{\prime}|\leq\|L\|^{k-1}(k-1)^{\frac{k-1}{2}}$. In case $L$ is a zero-one
matrix, Hadamard’s determinant bound is stronger. We have

$C\leq\frac{(k+1)^{\frac{k+1}{2}}}{2^{k}}\ \textrm{ and }\
c\geq\frac{2^{k-1}}{k^{\frac{k}{2}}}.$

We are able to strengthen Theorem 2.2 to the full asymptotic with a power
saving error term.

######  Theorem 3.1.

We have

$\sum_{\begin{subarray}{c}{a_{1},\ldots,a_{v}\leq x}\\\ {G\textrm{-wise
coprime}}\\\ {\forall_{i\leq k},\ \prod_{j\in A_{i}}a_{j}\leq
x}\end{subarray}}\frac{1}{a_{1}\cdots a_{v}}=P_{v}(\log
x)+O(x^{-\theta+\epsilon})$

where $P_{v}$ has degree $v$ with the leading coefficient
$\rho(G)\mathrm{vol}(D)$ in view of Theorem 2.2 and

$\theta=\frac{2^{k}}{(k+1)^{\frac{k+1}{2}}}\cdot\frac{3}{6k+\max(v-4,0)}.$

###### Proof.

This follows from Lemma 3.1 and Proposition 3.2. The arithmetic function

$f(n_{1},\ldots,n_{k})=\sum_{\begin{subarray}{c}{\forall_{r},a_{r}\in\mathbb{N}}\\\
{G\textrm{-wise coprime}}\\\ {\forall_{i\leq k},\ \prod_{j\in
A_{i}}a_{j}=n_{i}}\end{subarray}}\frac{1}{a_{1}\cdots a_{v}}$

in (13) satisfies the conditions of Proposition 3.2. We have
$\boldsymbol{\alpha}={\bf 0}$ so that $|J(\boldsymbol{\alpha})|=k$ and
$w=\ell=k$. Hence, the result follows. ∎

###  3.4 Proof of Theorem 1.1

(6) and (7) follow from the analysis of Section 3.2 and Theorem 3.1 with
$v=2^{k}-1$ and $v=2^{k}-2$ respectively. Thus, for $k\geq 3$, (6), (7), and
(8) hold with

$\theta^{(1)}_{k}=\theta^{(3)}_{k}=\frac{2^{k}}{(k+1)^{\frac{k+1}{2}}}\cdot\frac{3}{2^{k}+6k-5},\
\
\theta^{(2)}_{k}=\frac{2^{k}}{(k+1)^{\frac{k+1}{2}}}\cdot\frac{3}{2^{k}+6k-6}.$

We have $w=\ell=k$ in these two cases (6) and (7). The main term of (8) is
obtained by $\boldsymbol{\alpha}\cdot{\bf 1}=k$, $v=2^{k}-1$, $w=0$, $\ell=k$,
and Proposition 3.2.

By Section 3.2, the leading coefficients of $P^{(1)}_{2^{k}-1}$ and
$P^{(2)}_{2^{k}-2}$ are $c_{k}=\rho(G_{k})\mathrm{vol}(D_{k})$ and
$c_{k}^{(2)}=(2^{k}-1)c_{k}$ respectively. Computations of $\rho(G_{k})$ and
$\mathrm{vol}(D_{k})$ are continued on Section 4. The leading coefficient of
$P^{(3)}_{2^{k}-k-1}$ is positive in view of [K, Theorem 1.3] or [dlB, Theorem
2]. We have $\ell=k$ for all three cases (6), (7), and (8).

##  4 Computations of $\rho(G_{k})$ and $\mathrm{vol}(D_{k})$

###  4.1 The number $\rho(G_{k})$

In [HLT, (2.5)], it was proved that

$\sum_{n\leq x}\frac{\alpha_{k}(n)}{n}\leq S_{k}(x)$

where
$\alpha_{k}(n)=\sum_{\begin{subarray}{c}{n_{1},\ldots,n_{k}\in\mathbb{N}}\\\
{[n_{1},\ldots,n_{k}]=n}\end{subarray}}1$. Using $\alpha_{k}$ is
multiplicative and $\alpha_{k}(p)=2^{k}-1$ for each prime $p$, they applied
[HLT, Theorem 2.2] to prove

$\sum_{n\leq
x}\frac{\alpha_{k}(n)}{n}=\frac{C_{k}}{(2^{k}-1)!}\log^{2^{k}-1}x+O(\log^{2^{k}-2}x)$

where

$C_{k}=\prod_{p\in\mathcal{P}}\left(1-\frac{1}{p}\right)^{2^{k}-1}\sum_{\nu=0}^{\infty}\frac{(\nu+1)^{k}-\nu^{k}}{p^{\nu}}.$

In view of Theorem 2.2, we have

$\sum_{n\leq
x}\frac{\alpha_{k}(n)}{n}=\sum_{\begin{subarray}{c}{a_{1},\ldots,a_{v_{k}}\leq
x}\\\ {G_{k}\textrm{-wise coprime}}\\\ {a_{1}\cdots a_{v_{k}}\leq
x}\end{subarray}}\frac{1}{a_{1}\cdots
a_{v_{k}}}=\rho(G_{k})\textrm{vol}(T_{k})\log^{2^{k}-1}x+O(\log^{2^{k}-2}x)$

where

$T_{k}=\\{(u_{1},\ldots,u_{v_{k}})\in[0,\infty)^{v_{k}}\ |\
u_{1}+\cdots+u_{v_{k}}\leq 1\\}.$

Since $\textrm{vol}(T_{k})=\frac{1}{(2^{k}-1)!}$, we have

$\rho(G_{k})=C_{k}=\prod_{p\in\mathcal{P}}\left(1-\frac{1}{p}\right)^{2^{k}-1}\sum_{\nu=0}^{\infty}\frac{(\nu+1)^{k}-\nu^{k}}{p^{\nu}}.$

Thus, we are able to evaluate $\rho(G_{k})$ without knowing precise
information about $G_{k}$.

We have three expressions for $\rho(G_{k})$ from [AH], [H2], and [HLT]. These
suggest the identities for $|x|<1$,

$\sum_{F\subseteq
E_{k}}(-1)^{|F|}x^{v(F)}=\sum_{m=0}^{v_{k}}i_{m}(G_{k})\left(1-x\right)^{v_{k}-m}x^{m}=(1-x)^{2^{k}-1}\sum_{\nu=0}^{\infty}((\nu+1)^{k}-\nu^{k})x^{\nu}.$

Analyzing the proofs in these papers, the values at $1/p$ for each prime $p$
are identical. Thus, we see that the above identities are indeed true.

We may recover some information about $G_{k}$ by the last expression. For
example, we obtain the number of edges in $G_{k}$ by equating the coefficients
of $x^{2}$, thus

$-|E_{k}|=(3^{k}-2^{k})-(2^{k}-1)^{2}+\binom{2^{k}-1}{2}=3^{k}-2^{k-1}(2^{k}+1)$

so that $|E_{2}|=1$, $|E_{3}|=9$, and $|E_{4}|=55$. Also, we have

$\sum_{\nu=0}^{\infty}\nu^{k}x^{\nu}=\frac{1}{1-x}\sum_{m=1}^{k}S(k,m)m!\left(\frac{x}{1-x}\right)^{m}$

where $S(k,m)$ is the Stirling number of the second kind. Then

$\sum_{\nu=0}^{\infty}((\nu+1)^{k}-\nu^{k})x^{\nu}=\frac{1}{x}\sum_{m=1}^{k}S(k,m)m!\left(\frac{x}{1-x}\right)^{m}.$

By $\frac{1}{x}=1+\left(\frac{x}{1-x}\right)^{-1}$, we obtain $i_{m}(G_{k})$
in terms of the Stirling numbers. To see this, let $t=\frac{x}{1-x}$, we have

$\sum_{m=0}^{v_{k}}i_{m}(G_{k})\
t^{m}=\left(1+\frac{1}{t}\right)\sum_{m=1}^{k}S(k,m)m!\ t^{m}.$

Hence,

$i_{m}(G_{k})=S(k,m)m!+S(k,m+1)(m+1)!.$

Since the above vanishes when $m>k$, we have

$\sum_{F\subseteq
E_{k}}(-1)^{|F|}x^{v(F)}=\sum_{m=0}^{k}\big{(}S(k,m)m!+S(k,m+1)(m+1)!\big{)}\left(1-x\right)^{v_{k}-m}x^{m}.$

Therefore, we are able to write $f_{G_{k}}(p^{a})$ in terms of the Stirling
numbers and binomial coefficients. The following are the polynomials used in
the Euler product of $\rho(G_{k})$ obtained after expanding the right side.

$\displaystyle Q_{G_{2}}(x)$ $\displaystyle=1-x^{2},$ $\displaystyle
Q_{G_{3}}(x)$ $\displaystyle=1-9x^{2}+16x^{3}-9x^{4}+x^{6},$ $\displaystyle
Q_{G_{4}}(x)$
$\displaystyle=1-55x^{2}+320x^{3}-891x^{4}+1408x^{5}-1155x^{6}+1155x^{8}-1408x^{9}$
$\displaystyle+891x^{10}-320x^{11}+55x^{12}-x^{14}.$

###  4.2 The polytope $D_{k}^{*}$

As noted in Section 3.1, the volume of polytope defined by a finite set of
inequalities can be computed by a SageMath worksheet. Since
$\mathrm{vol}(D_{k})=\mathrm{vol}(D_{k}^{*})/(2^{k}-1)$, we use the following
code to generate a SageMath worksheet for $D_{k}^{*}$.

⬇

file2 = open(’Polytope.txt’, ’w+’)

k=input()

s=[[0 for y in range(2**k-1)] for x in range(k+2**k-2)]

for i in range(k): #Giving hyperbolic constraints.

s[i][0]=1

for j in range(2**k-1):

if j&(1<<i):

s[i][j]=-1 #When i-th bit from the right is 1, assign -1.

for z in range(k,k+2**k-2): #Restricting variables to be nonnegative.

s[z][z-k+1]=1

file2.write(’P=Polyhedron(ieqs=’)

file2.write(str(s))

file2.write(’)’)

file2.write(’\n’)

file2.write(’P.volume()’)

file2.close()

A python code generating a SageMath worksheet for $D_{k}^{*}$

The output in case $k=3$ is

⬇

P=Polyhedron(ieqs=[[1, -1, 0, -1, 0, -1, 0],

[1, 0, -1, -1, 0, 0, -1],

[1, 0, 0, 0, -1, -1, -1],

[0, 1, 0, 0, 0, 0, 0],

[0, 0, 1, 0, 0, 0, 0],

[0, 0, 0, 1, 0, 0, 0],

[0, 0, 0, 0, 1, 0, 0],

[0, 0, 0, 0, 0, 1, 0],

[0, 0, 0, 0, 0, 0, 1]])

P.volume()

The output in case $k=4$ is

⬇

P=Polyhedron(ieqs=[

[1, -1, 0, -1, 0, -1, 0, -1, 0, -1, 0, -1, 0, -1, 0],

[1, 0, -1, -1, 0, 0, -1, -1, 0, 0, -1, -1, 0, 0, -1],

[1, 0, 0, 0, -1, -1, -1, -1, 0, 0, 0, 0, -1, -1, -1],

[1, 0, 0, 0, 0, 0, 0, 0, -1, -1, -1, -1, -1, -1, -1],

[0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],

[0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],

[0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],

[0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],

[0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0],

[0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0],

[0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0],

[0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0],

[0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0],

[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0],

[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0],

[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0],

[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0],

[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1]])

P.volume()

Running these on SageMath, we obtain $\mathrm{vol}(D_{3}^{*})=11/480$ and
$\mathrm{vol}(D_{4}^{*})=739/25830604800$. Thus, $\mathrm{vol}(D_{3})=11/3360$
and $\mathrm{vol}(D_{4})=739/387459072000$.

###  4.3 The polytope $D_{k}^{**}$

Recall that

$D^{**}_{k}=\left\\{(t_{i})_{\begin{subarray}{c}{i\leq{v_{k}}}\\\ {i\neq
2^{t}}\end{subarray}}\in[0,\infty)^{2^{k}-k-1}\ \bigg{|}\forall_{i\leq k},\
\sum_{j\in A_{k,i}-\\{2^{i-1}\\}}t_{j}\leq 1\right\\}.$

The volume of $D^{**}$ is $1/(2^{k}-k-1)$ times the volume of

$D^{***}_{k}=\left\\{(t_{i})_{\begin{subarray}{c}{i\leq{v_{k}-1}}\\\ {i\neq
2^{t}}\end{subarray}}\in[0,\infty)^{2^{k}-k-2}\ \bigg{|}\forall_{i\leq k},\
\sum_{j\in A_{k,i}-\\{2^{i-1},v_{k}\\}}t_{j}\leq 1\right\\}.$

We use the following code to generate a SageMath worksheet for $D_{k}^{***}$.

⬇

file2 = open(’Polytope.txt’, ’w+’)

k=input()

s=[[0 for y in range(2**k-1)] for x in range(2*k+2**k-2)]

for i in range(k): #Giving hyperbolic constraints.

s[i][0]=1

for j in range(2**k-1):

if j&(1<<i):

s[i][j]=-1

#When i-th bit from the right is 1, assign -1.

for z in range(k,k+2**k-2): #Restricting variables to be nonnegative.

s[z][z-k+1]=1

for u in s:

for v in range(k):

del u[2**v-v] #Delete positions 2^j.

rows = [i for i in s if set(i)!={0}]

file2.write(’P=Polyhedron(ieqs=’)

file2.write(str(rows))

file2.write(’)’)

file2.write(’\n’)

file2.write(’P.volume()’)

file2.close()

A python code generating a SageMath worksheet for $D_{k}^{***}$

The output in case $k=3$ is

⬇

P=Polyhedron(ieqs=[[1, -1, -1, 0], [1, -1, 0, -1],

[1, 0, -1, -1], [0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]])

P.volume()

The output in case $k=4$ is

⬇

P=Polyhedron(ieqs=[

[1, -1, -1, 0, -1, -1, 0, -1, 0, -1, 0],

[1, -1, 0, -1, -1, 0, -1, -1, 0, 0, -1],

[1, 0, -1, -1, -1, 0, 0, 0, -1, -1, -1],

[1, 0, 0, 0, 0, -1, -1, -1, -1, -1, -1],

[0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0],

[0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0],

[0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0],

[0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0],

[0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0],

[0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0],

[0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0],

[0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0],

[0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0],

[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1]])

P.volume()

Running these on SageMath, we obtain
$\mathrm{vol}(D_{3}^{**})=\mathrm{vol}(D_{3}^{***})/4=1/16$ and
$\mathrm{vol}(D_{4}^{**})=\mathrm{vol}(D_{4}^{***})/11=299/479001600$.

## References

  * [AH] J. Arias-de-Reyna, R. Heyman, _Counting tuples restricted by pairwise coprimality conditions_ , J. Integer. Seq, 18 (2015), Article 15.10.4.
  * [BNP] M. Balazard, M. Naimi, Y. -F. S. Pétermann, _Étude d’une somme arithmétique multiple liée à la fonction de Möbius_ , Acta Arith. 132, (2008), pp. 245-298.
  * [dlB] R. de la Bretèche, _Estimation de sommes multiples de fonctions arithmétiques_ , Compos. Math. 128 (2001), pp. 261-298.
  * [EST] D. Essouabri, C. Salinas Zavala, L. Tóth, _Mean values of multivariable multiplicative functions and applications to the average number of cyclic subgroups and multivariable averages associated with the LCM function_ , J. Number Theory, in press, available at https://www.sciencedirect.com/science/article/pii/S0022314X2100281X
  * [FF] J. Fernández, P. Fernández, _On the probability distribution of the gcd and lcm of $r$-tuples of integers_, available at https://arxiv.org/abs/1305.0536
  * [H1] J. Hu, _The probability that random positive integers are $k$-wise relatively prime_, Int. J. Number Theory, 9. 5(2013), pp. 1263-1271.
  * [H2] J. Hu, _Pairwise relative primality of positive integers_ , available at https://arxiv.org/abs/1406.3113
  * [HT] T. Hilberdink, L. Tóth, _On the average value of the least common multiple of $k$ positive integers_, J. Number Theory, 169 (2016), pp. 327-341.
  * [HLT] T. Hilberdink, F. Luca, L. Tóth, _On certain sums concerning the gcd’s and lcm’s of $k$ positive integers_, Int. J. Number Theory, 16. no. 1(2020), pp. 77-90.
  * [K] S. Kim, _On the distribution of lcm of $k$-tuples and related problems_, submitted, available at https://arxiv.org/abs/2102.05480
  * [L] H. -Q. Liu, _On Euler’s function_ , P. Roy. Soc. Edinb. A, 146A (2016), pp 769-775.
  * [Ti] E. C. Titchmarsh, _The Theory of the Riemann Zeta-function 2nd Edition_ , Oxford Science Publications.
  * [To] L. Tóth, _The probability that $k$ positive integers are pairwise relatively prime_, Fibonacci Quart. 40 (2002), pp. 13-18.
  * [TZ] L. Tóth, W. Zhai, _On the error term concerning the number of subgroups of the groups $\mathbb{Z}_{m}\times\mathbb{Z}_{n}$ with $m,n\leq x$_, Acta Arith. 183, (2018), pp. 285-299.


# Multi-User Multi-Carrier Differential Chaos Shift Keying Communication
System

Georges Kaddoum, François-Dominique Richardson, Sarra Adouni, François Gagnon,
Claude Thibeault  Department of electrical engineer, LACIME Laboratory∗
Université du Québec, École de technologie supérieure
Montreal, Canada
E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

In this paper, a multi user Multi-Carrier Differential Chaos Shift Keying (MC-
DCSK) modulation is presented. The system endeavors to provide a good trade-
off between robustness, energy efficiency and high data rate, while still
being simple. In this architecture of MC-DCSK system, for each user, chaotic
reference sequence is transmitted over a predefined subcarrier frequency.
Multiple modulated data streams are transmitted over the remaining subcarriers
allocated for each user. This transmitter structure saves energy and increases
the spectral efficiency of the conventional DCSK system.

###### Index Terms:

Multiple access communication, Multi-carrier DCSK, Energy efficiency,
Performance analysis.

††publicationid: pubid: 978-1-4673-2480-9/13/$31.00 © 2013 IEEE
††footnotetext: * This work has been supported in part by Ultra Electronics
TCS and the Natural Science and Engineering Council of Canada as part of the
’High Performance Emergency and Tactical Wireless Communication Chair’ at
École de technologie supérieure.

##  I Introduction

The demand for wireless services is in constant rise. Multi-carrier (MC)
transmission, since it has the advantages of high spectral efficiency,
robustness to frequency selective fading, and feasibility of low-cost
transceiver implementation is a strong candidate for many wireless
applications. Several combinations of multi-carrier and Code Division Multiple
Access (CDMA), are proposed in the literature [1, 2]. In MC-CDMA, one-bit
chips are spread over $M$ subcarriers in the frequency domain [1], while for
MC-DS-CDMA, time and frequency spreading is used (i.e. TF-domain spreading)
[2].

The chaotic signals have been shown to be well suited for spread-spectrum
modulation because of their inherent wideband characteristic [3] [4] [5],
mitigation of fading channels, jamming resistance and low probability of
intercept (LPI) [6]. In addition, chaos-based sequences give good results as
compared to Gold and independent and identically distributed sequences for
reducing the peak-to-average power ratio (PAPR) [7].

A proposed system with a non-coherent receiver, named differential chaos shift
keying (DCSK) system, in which chaotic synchronization is not used on the
receiver side, delivers a good performance in multipath channels [8].
Furthermore, differential non-coherent systems are better suited than coherent
ones for time and frequency selective channels [9]. In the DCSK system, each
bit duration is divided into two equal slots. In the first slot, a reference
chaotic signal is sent. Depending on the bit being sent, the reference signal
is either repeated or multiplied by the factor $-1$ and transmitted in the
second slot. A significant drawback of DCSK is that for each bit one reference
and half the bit duration is spent sending non-information-bearing reference
samples [3]. This can be accounted as being energy-inefficient and a serious
data rate reducer. In [10], the spectral efficiency of the DCSK is improved,
but the system receiver requires an RF delay line, which is not easy to
implement because of the wide bandwidth involved. In a study to overcome the
problem of RF delay in DCSK systems, Xu et al. proposed a Code Shifted
Differential Chaos Shift Keying (CS-DCSK) system [11]. In their system, the
reference and the information bearing signals are separated by Walsh code
sequences, and then transmitted in the same time slot. For such systems, there
is no need for a delay line at the receiver end. An improved version of the
high spectral efficiency DCSK system by [11] is presented in [12], where
chaotic codes are used instead of Walsh codes, with different receiver
structures. Another design based on an ultra-wideband system using chaotic
signals for low complexity, low cost, low power, and low rate is presented in
[13].

In this paper, we first introduce a new design of a multi-user, multi-carrier
DCSK system (MC-DCSK). On the transmitter side, $M$ subcarriers are assigned
for each user, where one subcarrier is used to transmitting the references
slot, while the $M-1$ other frequencies will carry the transmitted bits. The
proposed system solves the RF delay line problem mentioned in [11], provides
from the properties of DCSK system in terms of resistance to interference,
increases the data rate, and optimizes the transmitted energy of the DCSK
system with a simple transmitter/receiver design. The analytical performance
derivation of DCSK communication system is studied in [14] [15] [16] [17] and
the transmission security is improved in [18]. In this paper, for the space
available, we concentrate our efforts to explain the proposed system design,
where the analytical derivation and cognitive multiple access techniques of
MC-DCSK system will be studied in future work.

In this paper, section II describes the characteristics of chaos-based systems
with an emphasis on DCSK. Section III covers the architecture of the multi-
user MC-DCSK system. The energy of the system is examined in section IV.
Simulation results and discussions are presented in section V, and concluding
remarks are presented in section VI.

Figure 1: Block diagram of the single-user MC-DCSK system

##  II Drawback of DCSK communication scheme

In DCSK modulator, each bit $s_{i}~{}=~{}\\{-1,\>+1\\}$ is represented by two
sets of chaotic signal samples, with the first set representing the reference,
and the second carrying data. If $+1$ is transmitted, the data-bearing
sequence is equal to the reference sequence, and if $-1$ is transmitted, an
inverted version of the reference sequence is used as the data-bearing
sequence. Let $2\beta$ be the spreading factor, defined as the number of
chaotic samples sent for each bit, where $\beta$ is an integer. During the
$i^{th}$ bit duration, the output of the transmitter $e_{i,k}$ is

$e_{i,k}=\left\\{\begin{array}[]{l}x_{i,k}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\rm
for}\,\,1<k\leq\beta,\\\ s_{i}x_{i,k-\beta\,\,\,}\,\,\,\,{\rm
for}\,\,\beta<k\leq 2\beta,\\\ \end{array}\right.$ (1)

where $x_{k}$ is the chaotic sequence used as reference and $x_{k-\beta}$ is
the delayed version of the reference sequence.

In this system, half the bit duration time is spent sending a non-information-
bearing reference. Therefore, the data rate of this architecture is seriously
reduced compared to other systems using the same bandwidth, leading to a loss
of energy. The reference sequence dissipates half the energy of each bit.

##  III Multi-User Multi-carrier DCSK system

The system’s architecture is intended to be of low complexity thus simple to
implement. Numerous extensions could be performed to this system for different
performance optimizations.

###  III-A The transmitter

The MC-DCSK system benefits from the non-coherent advantages of DCSK and the
spectral efficiency of multi-carrier modulation. Here, we consider an MC-DCSK
system using discrete chaotic sequence values for modulation and square-root-
raised-cosine filter chip waveforms. For mathematical simplification, the
equations describe the MC-DCSK system for one user. As shown in Figure 1, for
each user, a reference chaotic code $x_{u}$ is generated to be used as a
reference and spreading code. The input information sequence is first
converted into $U$ parallel data sequences $s_{u}(t)$ for $u=1,\>2,...U$. The
independent data sequence $s_{u}(t)$ with equal probability value is $+1,$ or
$-1$, where

$s_{u}(t)=\sum\limits_{i=1}^{M-1}{s_{u,i}}(t)\cdot$ (2)

After a serial-to-parallel conversion, the $u^{th}$ ${M-1}$ bits substream is
spread due to multiplication in time with the same chaotic spreading code
$x_{u}$.

$x_{u}(t)=\sum\limits_{k=1}^{\beta}{x_{u,k}}h(t-kT_{c}),$ (3)

where $h(t)$ is the square-root-raised-cosine filter. This filter is band-
limited and is normalized to have unit energy. Let
$H(f)=\textit{F}\left\\{{h(t)}\right\\}$, where F denotes a Fourier transform.
It is assumed that $H(f)$ is limited to $[-B_{c}/2,\>B_{c}/2]$ which satisfies
the Nyquist criterion with a rolloff factor $\alpha$ ($0\leq\alpha\leq 1$)
where $B_{c}=(1+\alpha)/T_{c}$.

The $x_{u}$ modulates the subcarrier assigned to transmit the reference
signal, after which the data signals spread by ${M-1}$ modulate the ${M-1}$
subcarriers. Therefore, the transmitted signal of the mono-user MC-DCSK is
given by:

$\begin{array}[]{l}e(t)={x_{u}(t)\cos(2\pi f_{1}t+\phi_{1})}+\\\
\sum\limits_{i=2}^{M}{{s_{u,i}(t)x_{u}(t)\cos(2\pi f_{i}t+\phi_{i})}}\\\
\end{array},$ (4)

where $\phi_{i}$ represents the phase angle introduced in the carrier
modulation process. In this paper, we normalize the transmitted energy in
every subcarrier.

For the MC-DCSK, the frequency corresponding to the $i^{t}h$ subcarrier is
$f_{i}=f_{p}+i/T_{c}$, where $f_{p}$ is the fundamental carrier frequency. The
minimum spacing between two adjacent subcarriers equals
$\Delta=(1+\alpha)/T_{c}$, which is a widely used assumption.

Figure 2: The power spectral density of a band-limited multi-user MC-DCSK
system.

Figure 2 shows the power spectral density (PSD) of the multi-user MC-DCSK
system. Let $B$ be the bandwidth for each user. The maximal number of users
using the channel in the same time is determined from the total allocated
bandwidth $B_{t}$ for this system determine. When both $T_{b}$ and $B$ are
set, the chip duration $T_{c}$ as well as the spreading factor $\beta$ depend
on the number of carrier $M$, the bandwidth $B_{c}$ of each subchannel or the
carrier spacing $\Delta$. In our design, we divide the total band $B$ into $M$
equi-width frequency bands, as shown in Figure 2, where all bands are
disjoint. The bandwidth of each carrier band $B_{c}$ is:

$B_{c}=(1+\alpha)/T_{c};\>where\>B=MB_{c},$

Thus, the spreading factor $\beta=T_{b}/T_{c}$ function of the system
parameters is :

$\beta=\frac{{T_{b}B}}{{M(1+\alpha)}}\cdot$ (5)

In order to validate our system design and evaluate its parameters, such as
the spreading factor, the bit energy, and the bit error rate performance, we
assume that our channel is AWGN. In addition we assume that no interference
between subcarriers. In this case we evaluate the received signal for one
user.

$r(t)=v(t)+n(t),$ (6)

where $r(t)$ is the received signal, $n(t)$ is an AWGN noise with zero mean
and power spectral density of $N_{0}/2$.

###  III-B The receiver

The block diagram of the MC-DCSK receiver is illustrated in Figure 1. One of
the objectives of this design was to provide a simple, easy-to-implement
receiver providing good performance. We consider a set of matched filters,
each demodulating the desired signal of the corresponding carrier frequency
$f_{i}$, and then the signals are sampled every $kT_{c}$ time. The outputs
discrete signals are stored in matrix memory. The matrix implementation of the
receiver simplifies the parallel data recovery, where the decoding algorithm
is:

First, at the same time, the output of the first match is stored in matrix $P$
and the ${M-1}$ data signals are stored in the second matrix $S$, where:

$P=(x_{u,1}+n_{u,1},\>x_{u,2}+n_{u,2},...\>x_{u,\beta}+n_{u,\beta}),$

where $n_{u,k}$ is the $k^{th}$ sample of additive Gaussian noise added to the
reference signal.

The matrix $S$ is:

$S=\left({\begin{array}[]{*{20}c}{s_{u,1}x_{u,1}+n^{(1)}_{u,1}}&\ldots&{s_{u,1}x_{u,\beta}+n^{(1)}_{u,\beta}}\\\
\vdots&\vdots&\vdots\\\
{s_{u,M-1}x_{u,1}+n^{(M-1)}_{u,1}}&\ldots&{s_{u,M-1}x_{u,\beta}+n^{(M-1)}_{u,\beta}}\\\
\end{array}}\right)\cdot$

where $n^{(i)}_{u,k}$ is the $k^{th}$ sample of additive Gaussian noise added
to the $i^{th}$ bit.

Finally, after $\beta$ clock cycles, all the samples are stored, and the
decoding step is activated. The transmitted ${M-1}$ bits are recovered in
parallel by computing the sign of the resultant vector of the matrix product:

$\hat{s}_{u}=sign(P\times S^{\prime})\cdot$ (7)

where $\times$ is the matrix product and ′ is the matrix transpose operator.
In fact, this matrix product can be seen as a set of a parallel correlator
where the reference signal multiplies each data slot, and the result is summed
over the duration $\beta Tc$.

##  IV Energy efficiency

The energy efficiency of the proposed system is improved as compared to the
DCSK system. In fact, for the DCSK system, a new chaotic reference is
generated for every transmitted bit, and in our case, one reference is shared
with ${M-1}$ modulated bits. For a conventional DCSK system, the transmitted
bit energy $E_{b}=E_{data}+E_{ref}$, where $E_{data}$ and $E_{ref}$ are the
energies to transmit the data and reference respectively. Without loss of
generality, the data and the reference energies are equal
$E_{data}=E_{ref}=T_{c}\sum\limits_{k=1}^{\beta}{x_{i,k}^{2}}$.

Then for DCSK system, the transmitted energy for a given bit $i$ is
$E_{b}=2T_{c}\sum\limits_{k=1}^{\beta}{x_{i,k}^{2}}$.

In the MC-DCSK system, the bit energy is the sum of the data carrier energy
and a part of the reference energy:

$E_{b}=E_{data}+\frac{{E_{ref}}}{{M-1}}\cdot$ (8)

In our system, the energies on the $M$ subcarriers are equal:

$E_{data}=E_{ref}=T_{c}\sum\limits_{k=1}^{\beta}{x_{i,k}^{2}}.$ (9)

To study the energy efficiency, we compute the transmitted Data-energy-to-Bit-
energy Ratio (DBR):

$DBR=\frac{E_{data}}{E_{b}}\cdot$ (10)

In a conventional DCSK system, half the energy is transmitted into the
reference to achieve modulation for each bit, and then the DBR is equal
$DBR=\frac{1}{2}$.

In the MC-DCSK system the DBR is :

$DBR=\frac{M-1}{M}\cdot$ (11)

Figure 3: DBR for a system for various amount of data subcarriers.

As shown in Figure 3, for $M=2$ where we have one reference for one bit, in
this case, the MC-DCSK system is equivalent to a DCSK system with
$DBR=\frac{1}{2}$. This means that $50\%$ of the energy is used from the total
bit energy to transmit the reference. We can see clearly that for $M>20$, the
reference energy accounts for less than $5\%$ of the total bit energy $E_{b}$.

##  V Simulation results and discussions

Since the space between subcarriers guarantee a free interferences, the
performance in mono-user or in multi user case in AWGN channel remains the
same. The parameters of the simulation are set as follows: the MC-DCSK system
uses the square-root-raised-cosine chip waveform for a roll-off factor
$\alpha$ equal to $0.25$. As shown in equation (5), the spreading factor is
computed as a function of the number of subcarriers $M$, the bit duration
$T_{b}$, and the total allocated bandwidth $B$. In our simulations, we set the
bit duration $T_{b}=400$, $B=1$ and for $M=64$ the allowed spreading factor
$\beta=5$, for $M=16$ subcarriers $\beta=20$, for $M=8$ $\beta=40$, and for
$M=2$ $\beta=160$.

Figure 4: BER performance of MC-DCSK for different number of subcarriers and
spreading factors

Figure 4 presents the performances obtained from the Monte Carlo simulations
of the MC-DCSK system for different spreading factors and number of
subcarriers $M$ per user. In Figure 5 we study the effect of the number of
subcarriers on the system performance. To that end, we set the spreading
factor to $\beta=5$ and the bit duration $T_{b}$, and then we assume that the
bandwidth $B$ is wide enough to support any number of subcarriers $M$. Figure
5 shows interesting results of our proposed MC-DCSK system in terms of
performance enhancement. In fact, for a given spreading factor, when the
number of subcarriers $M$ increases, the $DBR$ ratio tends toward one, meaning
that less energy is used to transmit the reference code. This performance
means that for high number of subcarriers $M$, we need less energy to reach a
given BER. In this figure, we show the performance improvement by simulation
for $M=2$ and $M=64$, with a fixed spreading factor equal to $\beta=5$. In
this case, this result compares the performance of the proposed system with
that of the conventional DCSK. In fact, when $M=2$, the MC-DCSK system is
equivalent to a DCSK system because one reference is required to transmit
every bit. In the same figure, we can observe a degradation in performance
between the MC-DCSK system for $M=64$ and the coherent BPSK one. This
degradation comes from the two noise sources added to the reference and the
data carrier signals.

Figure 5: BER comparison of MC-DCSK for M=64 and DCSK where the spreading
factor $\beta=5$

##  VI Conclusion

An energy-efficient non-coherent multi-carrier spread spectrum system has been
presented. From the outstanding energy inefficiency drawback imposed by time-
multiplexed differential modulations, a novel frequency multiplexed
architecture is designed. The multi-carrier characteristic of this novel
design enables significant energy savings and a higher spectral efficiency as
compared to differential systems because for each user, the reference signal
is only sent once for $M-1$ parallel bits. The energy efficiency of the
proposed system is analyzed and a $DBR$ is derived, with results showing that
for $M>20$ subcarriers, the energy lost in transmitting the reference is less
than $5\%$ of the total bit energy. To compare the performance of the proposed
system with that of the DCSK, the simulated BERs are plotted with the same
spreading factor, where results prove an increase in performance as compared
to the conventional DCSK. Our future work will focus on defining new cognitive
multi-user access strategies and performance improvement of this system.

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