1 Introduction
For any
$n\in \mathbb {N}$
, let
$\Omega (n)$
be the number of prime divisors of n counted with multiplicity. Let
$\lambda (n)=(-1)^{\Omega (n)}$
be the Liouville function. It is well known (see, for example, [Reference Landau5]) that the prime number theorem (PNT) is equivalent to the assertion that
$$ \begin{align} \lim_{N \to \infty} \frac1N\sum_{n=1}^N \lambda(n)=0. \end{align} $$
In 2022, Bergelson and Richter [Reference Bergelson and Richter1] established a new dynamical generalisation of the PNT by placing (1.1) in a dynamical framework, showing that given any uniquely ergodic topological dynamical system
$(X, \mu , T)$
,
$$ \begin{align} \lim_{N\to\infty}\frac1N\sum_{n=1}^N f(T^{\Omega(n)}x)=\int_X f \,d\mu \end{align} $$
holds for any
$f\in C(X)$
and
$x\in X$
, where
$C(X)$
denotes the set of continuous functions defined on X. The equivalent form (1.1) of the PNT is recovered from (1.2) by taking
$(X,T)$
as a rotation on two points.
The Bergelson–Richter theorem (1.2) is a dynamical generalisation of some other number theoretic results. For example, taking
$(X,T)$
as a rotation on m points (
$m\ge 2$
) in (1.2) recovers a classical result of Pillai and Selberg [Reference Pillai8, Reference Selberg9], which states that
$\Omega (n)$
distributes evenly over all residue classes modulo m. Moreover, taking
$(X,T)$
as irrational circle rotations in (1.2) implies that the sequence
$\{\Omega (n)\alpha \}_{n\in \mathbb {N}}$
is uniformly distributed modulo
$1$
for any irrational number
$\alpha $
, a result mentioned by Erdős in [Reference Erdős3] without proof and later proved by Delange [Reference Delange2]. We refer to [Reference Bergelson and Richter1] for more interesting applications of (1.2).
In [Reference Bergelson and Richter1], Bergelson and Richter also established the following dynamical generalisation of the PNT for arithmetic progressions.
Theorem 1.1 [Reference Bergelson and Richter1, Corollary 1.16].
Let
$(X, \mu , T)$
be a uniquely ergodic topological dynamical system. Then,
$$ \begin{align} \lim_{N\to\infty}\frac1N \sum_{1\leq n \leq N} f(T^{\Omega(mn+r)}x)= \int_X f \,d\mu \end{align} $$
for any
$f\in C(X)$
,
$x\in X$
,
$m\in \mathbb {N}$
and
$r\in \{0,1\ldots ,m-1\}$
.
Indeed, taking rotation on two points in (1.3) gives
$$ \begin{align*} \lim_{N \to \infty} \frac1N\sum_{n=1}^N \lambda(mn+r)=0 \quad\mbox{for all } m\in \mathbb{N} \mbox{ and } r\in \{0,1\ldots,m-1\}, \end{align*} $$
which is an equivalent form of the PNT along arithmetic progressions (see [Reference Shapiro10]).
Let
$k\ge 2$
be an integer. A natural number n is said to be k-free if it has no divisor
$d>1$
which is a perfect kth power. We define
$\mu _k(n)=1$
if n is k-free and zero otherwise. By Theorem 1.1, the orbits
$\{f(T^{\Omega (mn+r)}x)\}_{n\in \mathbb {N}}$
are equi-distributed in the space X for any
$x\in X$
. In this note, we show that the orbits
$\{f(T^{\Omega (mn+r)}x)\}_{n\in \mathbb {N}}$
are also equi-distributed along a system of k-free numbers.
2 Main results
Let
$h, q\ge 1$
be integers. Let
$l_1,\ldots , l_h$
be nonnegative integers. In [Reference Mirsky7], Mirsky showed that the numbers
$qn$
such that
$qn+l_1, \ldots , qn+l_h$
are k-free have a natural density
$A_q(l_1,\ldots , l_h)$
defined by
$$ \begin{align} A_q(l_1,\ldots,l_h)=\frac1q\prod_{p^k\nmid q}\bigg(1-\frac{(p^k,q)\nu_q(p)}{p^k}\bigg), \end{align} $$
and
$\nu _q(p)=\#\{1\leq n\leq p^k: (p^k,q)\mid n, n\equiv l_i\ \ (\operatorname {mod}p^k) \text { for some } i, 1\leq i \leq h\}$
. The following is our main theorem, which shows that the Bergelson–Richter theorem holds for the numbers studied by Mirsky.
Theorem 2.1. Let
$h, q\ge 1$
be integers. Let
$l_1,\ldots , l_h$
be nonnegative integers. Let
$(X, \mu , T)$
be a uniquely ergodic topological dynamical system. Then,
$$ \begin{align} \lim_{N\to\infty}\frac1N \sum_{1\leq qn \leq N} \mu_k(qn+l_1)\cdots \mu_k(qn+l_h)f(T^{\Omega(mn+r)}x)= A_q(l_1,\ldots,l_h) \int_X f \,d\mu \end{align} $$
for any
$f\in C(X)$
,
$x\in X$
,
$m\in \mathbb {N}$
and
$r\in \{0,1\ldots ,m-1\}$
, where
$A_q(l_1,\ldots ,l_h)$
is defined by (2.1).
By Theorem 2.1, the orbits
$\{f(T^{\Omega (mn+r)}x)\}_{n\in \mathbb {N}}$
are equi-distributed along the numbers n such that
$qn+l_1, \ldots , qn+l_h$
are k-free for any
$x\in X$
. As a corollary, we obtain the following special case by taking
$q=1,m=1, r=0$
in (2.2).
Corollary 2.2. Let
$h\ge 1$
be an integer. Let
$l_1,\ldots , l_h$
be nonnegative integers. Let
$(X, \mu , T)$
be a uniquely ergodic topological dynamical system. Then,
$$ \begin{align*} \lim_{N\to\infty}\frac1N \sum_{1\leq n \leq N} \mu_k(n+l_1)\cdots \mu_k(n+l_h)f(T^{\Omega(n)}x)= A(l_1,\ldots,l_h) \int_X f \,d\mu \end{align*} $$
for any
$f\in C(X)$
and
$x\in X$
, where
$$ \begin{align*} A(l_1,\ldots,l_h) =\prod_{p} \bigg(1-\frac{\nu(p)}{p^k}\bigg) \end{align*} $$
and
$\nu (p)=\#\{l_i \ \ (\operatorname {mod}p^k): 1\leq i\leq h\}$
.
In [Reference Mirsky6], Mirsky showed that the numbers n such that
$n+l_1,\ldots , n+l_h$
are k-free have a natural density
$A(l_1,\ldots ,l_h)$
. Moreover, he obtained an asymptotic formula for the density with a better error term than [Reference Mirsky7]. By taking
$k=2, h=1$
and
$l_1=0$
in Corollary 2.2, we obtain the result in [Reference Bergelson and Richter1, Corollary 1.8]: for any uniquely ergodic topological dynamical system
$(X, \mu , T)$
,
$$ \begin{align} \lim_{N \to \infty} \frac1N\sum_{\substack{1\le n \le N\\ n\,\text{squarefree}}} f(T^{\Omega(n)}x)=\frac6{\pi^2}\int_X f \,d\mu \end{align} $$
for any
$f\in C(X)$
and
$x\in X$
. Taking
$(X,T)$
as a rotation on two points in (2.3) gives another well-known equivalent form of the PNT:
$$ \begin{align*} \lim_{N \to \infty} \frac1N\sum_{n=1}^N \mu(n)=0, \end{align*} $$
where
$\mu (n)$
is the Möbius function, which is defined to be
$1$
if
$n = 1$
,
$(-1)^k$
if n is the product of k distinct primes and zero otherwise. Therefore, Corollary 2.2 gives a dynamical extension of the PNT. Moreover, we obtain the following variant of the Bergelson–Richter theorem along twins of squarefree numbers by taking
$k=h=2, l_1=0, l_2=1$
in Corollary 2.2.
Corollary 2.3. Let
$(X, \mu , T)$
be a uniquely ergodic topological dynamical system. Then,
$$ \begin{align*} \lim_{N\to\infty}\frac1N \sum_{\substack{1\leq n \leq N\\ n,\, n+1 \text{ squarefree}}} f(T^{\Omega(n)}x)= \prod_{p}\bigg(1-\frac{2}{p^2}\bigg) \int_X f \,d\mu \end{align*} $$
for any
$f\in C(X)$
and
$x\in X$
.
Remark 2.4. In [Reference Mirsky7], Mirsky also studied the following three systems of positive integers relating to k-free numbers:
-
(1) positive integers n such that
$n+a_1,\ldots , n+a_l$
are k-free while
$n+b_1,\ldots , n+b_m$
are not, where
$a_1, \ldots , a_l, b_1,\ldots , b_m$
are distinct positive integers; -
(2) blocks of
$s\ k$
-free numbers,
$s\ge 1$
; -
(3) blocks of s numbers that are not k-free,
$s\ge 1$
.
Similar to the arguments in [Reference Mirsky7], one can apply Corollary 2.2 to establish the variants of the Bergelson–Richter theorem along these three systems of numbers.
In the next section, we will prove Theorem 2.1 by applying some ideas of Mirsky in [Reference Mirsky6, Reference Mirsky7], Hall in [Reference Hall4, Theorem 1], and Bergelson and Richter in [Reference Bergelson and Richter1, Corollary 1.8].
3 Proof of Theorem 2.1
To prove Theorem 2.1, we first cite some lemmas in [Reference Hall4, Reference Mirsky7]. For positive integers
$m_1,\ldots ,m_s$
and
$a_1,\ldots , a_s$
, it is well known that the system of congruences in n,
$$ \begin{align} n+a_i\equiv 0 \ \ (\operatorname{mod}m_i), \quad1\leq i \leq s, \end{align} $$
is soluble if and only if
$(m_i,m_j) \mid (a_i-a_j)$
for
$1\leq i<j\leq s$
. If (3.1) is soluble, then the solutions form exactly one residue class modulo
$[m_1,\ldots , m_s]$
. Let
$$ \begin{align*}E\bigg(\begin{matrix} m_1,\ldots, m_s \\ a_1,\ldots, a_s \end{matrix} \bigg)=1 \text{ or } 0\end{align*} $$
according to whether (3.1) is or is not soluble.
Lemma 3.1 [Reference Mirsky7, Lemma 3].
Let
$m_1,\ldots ,m_s, a_1,\ldots , a_s, q$
be given positive integers, and T the number of positive integers n such that
$$ \begin{align*} qn+a_i\equiv0 \ \ (\operatorname{mod}m_i),\quad 1\leq i \leq s \quad \text{and} \quad qn\leq N. \end{align*} $$
Then,
$$ \begin{align*} T=N\frac{E\bigg(\begin{matrix}q, m_1,\ldots, m_s \\ 0, a_1,\ldots, a_s \end{matrix} \bigg)}{[q,m_1,\ldots,m_s]} +O(1).\end{align*} $$
Lemma 3.2 [Reference Mirsky7, Lemma 6].
We have
$$ \begin{align*} \sum_{d_1=1}^\infty\cdots\sum_{d_h=1}^\infty \frac{\mu(d_1)\cdots \mu(d_h)}{[q,d_1^k,\ldots,d_h^k]} E\bigg(\begin{matrix}q, d_1^k,\ldots,d_h^k \\ 0, l_1,\ldots,l_h \end{matrix} \bigg) = \frac1q\prod_{p^k\nmid q}\bigg(1-\frac{(p^k,q)\nu_q(p)}{p^k}\bigg), \end{align*} $$
where
$\nu _q(p)=\#\{1\leq n\leq p^k: (p^k,q)\mid n, n\equiv l_i\ \ (\operatorname {mod}p^k) \text { for some } i, 1\leq i \leq h\}$
.
Lemma 3.3 [Reference Hall4, Lemma 2].
For each fixed h and
$\sigma>1$
,
$$ \begin{align*} \sum_{d_1>H}\cdots\sum_{d_h>H}\frac1{[d_1,\ldots, d_h]^\sigma} \ll_{h,\sigma} H^{1-\sigma}. \end{align*} $$
Now, we prove Theorem 2.1 by an elementary method.
Proof of Theorem 2.1.
Put
$a(n)=f(T^{\Omega (mn+r)}x)$
and
$\alpha =\int _X f \,d\mu $
. By Theorem 1.1,
$$ \begin{align} \lim_{N\to\infty}\frac1N \sum_{1\leq n \leq N} a(dn+t)=\alpha \end{align} $$
for any
$d\in \mathbb {N}$
and
$t\in \{0,1\ldots ,d-1\}$
. Clearly,
$a(n)$
is bounded. We may assume that
$|a(n)|\leq 1$
for all n. It is well known that
$\mu _k(n)=\sum _{d^k\mid n}\mu (d)$
. Let
$$ \begin{align*}\mu_{k,H}(n)= \sum_{\substack{d^k\mid n\\ d\leq H}} \mu (d) \quad\text{and}\quad e_{k,H}(n)=\sum_{\substack{d^k\mid n\\ d> H}} \mu (d)\end{align*} $$
for
$H\ge 2$
. Then,
$\mu _k=\mu _{k,H}+e_{k,H}$
. Define
$$ \begin{align*}S_H(N)= \sum_{1\leq qn\leq N} \mu_{k,H}(qn+l_1)\cdots \mu_{k,H}(qn+l_h) a(n).\end{align*} $$
By using
$\mu _k=\mu _{k,H}+e_{k,H}$
and expanding the left-hand side of (2.2),
$$ \begin{align} & \bigg|\!\sum_{1\leq qn \leq N} \mu_k(qn+l_1)\cdots \mu_k(qn+l_h)a(n)- S_H(N) \bigg| \notag \\ & \leq \sum_{1\leq i \leq h} \sum_{qn\leq N} |e_{k,H}(qn+l_i)| +\sum_{1\leq i,j \leq h} \sum_{qn\leq N} |e_{k,H}(qn+l_i)e_{k,H}(qn+l_j)| +\cdots \notag \\ &\quad +\sum_{qn\leq N} |e_{k,H}(qn+l_1)\cdots e_{k,H}(qn+l_h)|. \end{align} $$
Let
$l=\max \{l_i: 1\leq i \leq h\}$
. For
$1\leq r \leq h$
and
$1\leq i_1,\ldots , i_r \leq h$
, by Hölder’s inequality,
$$ \begin{align*} \sum_{qn\leq N} |e_{k,H}(qn+l_{i_1})\cdots e_{k,H}(qn+l_{i_r})|\leq \prod_{1\leq j \leq r} \bigg(\sum_{qn\leq N} |e_{k,H}(qn+l_{i_j})|^r\bigg)^{1/r} \leq \sum_{n\leq N+l} |e_{k,H}(n)|^r. \end{align*} $$
Moreover, by Lemma 3.3,
$$ \begin{align*} \sum_{n\leq N+l} |e_{k,H}(n)|^r \leq \sum_{n\leq N+l} \sum_{\substack{d_1^k\mid n \\ d_1> H}} \cdots \sum_{\substack{d_r^k\mid n\\ d_r> H}}1 \leq \sum_{d_1>H} \cdots \sum_{d_r>H}\frac{N+l}{[d_1,\ldots,d_r]^k} \ll \frac{N+l}{H^{k-1}}. \end{align*} $$
It follows that the right-hand side of (3.3) is bounded by
$$ \begin{align*} \sum_{r=1}^h\binom{h}{r} \frac{N+l}{H^{k-1}}=(2^h-1) \frac{N+l}{H^{k-1}}. \end{align*} $$
This implies that
$$ \begin{align} \sum_{1\leq qn \leq N} \mu_k(qn+l_1)\cdots \mu_k(qn+l_h)a(n)= S_H(N) + O_{h,k,l}\bigg(\frac{N}{H^{k-1}}\bigg). \end{align} $$
As regards
$S_H(N)$
, by the definition of
$\mu _{k,H}$
,
$$ \begin{align*}S_H(N)= \sum_{d_1\leq H}\cdots\sum_{d_h\leq H} \mu(d_1)\cdots \mu(d_h) \sum_{\substack{ qn\leq N \\ d_i^k \mid n+l_i \,{\scriptsize \text{for all }} i}}a(n).\end{align*} $$
Let
$T_q(N)$
be the number of positive integers n such that
$qn\leq N$
and
$d_i^k \mid n+l_i$
for all i with
$1\leq i\leq h$
. Then, by Lemma 3.1,
$$ \begin{align*}T_q(N)=N\frac{E\bigg(\begin{matrix}q, d_1^k,\ldots,d_h^k \\ 0, l_1,\ldots,l_h \end{matrix} \bigg)}{[q,d_1^k,\ldots,d_h^k]} +O(1).\end{align*} $$
If
$E(\begin {smallmatrix}q, d_1^k,\ldots ,d_h^k \\ 0, l_1,\ldots ,l_h \end {smallmatrix} )=1$
, we write
$n=q'm+t$
for some t with
$0\leq t<q'$
, where
$qq'=[q,d_1^k,\ldots ,d_h^k]$
.
If B is a finite nonempty set, we define
for any function
$b:B\to \mathbb {C}$
on B. Then,
for
$2\leq H\leq N$
.
Now, we fix H first. By (3.2), taking
$N\to \infty $
in (3.6),
$$ \begin{align} &\lim_{N\to\infty}\frac1N \sum_{1\leq qn \leq N} \mu_k(qn+l_1)\cdots \mu_k(qn+l_h)a(n) \notag\\ &\quad= \alpha \sum_{d_1\leq H}\cdots\sum_{d_h\leq H} \frac{\mu(d_1)\cdots \mu(d_h)}{[q,d_1^k,\ldots,d_h^k]} E\bigg(\begin{matrix}q, d_1^k,\ldots,d_h^k \\ 0, l_1,\ldots,l_h \end{matrix} \bigg) + O\bigg(\frac{1}{H^{k-1}}\bigg). \end{align} $$
Then, taking
$H\to \infty $
in (3.7) gives
$$ \begin{align*} &\lim_{N\to\infty}\frac1N \sum_{1\leq n \leq N} \mu_k(n+l_1)\cdots \mu_k(n+l_h)a(n)\\ &\quad= \alpha \sum_{d_1 =1}^\infty \cdots\sum_{d_h =1}^\infty \frac{\mu(d_1)\cdots \mu(d_h)}{[q,d_1^k,\ldots,d_h^k]} E\bigg(\begin{matrix}q, d_1^k,\ldots,d_h^k \\ 0, l_1,\ldots,l_h \end{matrix} \bigg) , \end{align*} $$
which is equal to
$A_q(l_1,\ldots , l_h) \alpha $
by Lemma 3.2. This completes the proof of Theorem 2.1.
Remark 3.4. In Theorem 1.1, the rate of convergence in (1.3) is
$o_{N\to \infty }(1)$
. If one can show a better bound than
$o_{N\to \infty }(1)$
in Theorem 1.1, then one can also obtain better bounds on the rate of convergence in Theorem 2.1 and Corollary 2.3 by using (3.6) and the arguments similar to [Reference Hall4, Theorem 1].
Acknowledgements
The author would like to thank Prof. Zhengxing Lian for the discussions on the number of consecutive squarefree numbers. The author is also grateful to the referee for a careful review and for the remarks which improved the previous version of the paper.
