1 Introduction
Given a large integer n, determining the size of its distinct prime factors as well as their relative sizes has been the focus of several studies over the past 75 years. For instance, if we let $\omega (n)$ stand for the number of distinct prime divisors of an integer $n\ge 2$ and
be these prime divisors, Erdoős [Reference Erdős6] proved in 1946 that, letting $\xi (n)\to \infty $ as $n\to \infty $ and given any small number $\varepsilon>0$ , then
thereby providing bounds on the size of the kth prime factor for most integers.
Thirty years later, Galambos [Reference Galambos7] provided important information on the relative size of the consecutive prime factors of an integer by showing that, given any small $\varepsilon>0$ and a function $j=j(x)$ which tends to infinity with x in such a manner that $j(x)\le (1-\varepsilon )\log \log x$ , then, for any fixed real number $z>1$ ,
thereby providing bounds on the gaps between the successive prime factors of most integers. Further refinements of these results were provided by many others, such as De Koninck and Galambos [Reference De Koninck and Galambos1] in 1987, and Granville [Reference Granville8, Reference Granville9] in 2007.
More recently, in [Reference De Koninck and Kátai3, Reference De Koninck and Kátai4], we further explored this topic by examining the spacings between those consecutive prime factors of an integer which are ‘close’ to one another, in the following sense. Given a real number $\lambda \in (0,1]$ , we introduced the arithmetic function
and proved (see Theorem 1 in [Reference De Koninck and Kátai3]) that
and that, for any given $\varepsilon>0$ ,
Here, we examine the spacings between particular types of prime divisors of n, such as prime divisors in certain congruence classes of primes and various other subsets of the set of prime numbers.
2 Setting the table
Let $\wp $ be the set of all primes. From here on, unless indicated otherwise, the letters p, q and $\pi $ will stand for primes, whereas $\pi (x)$ will stand for the number of primes not exceeding x. Also, we will be using the logarithmic integral ${\mbox {li}(x):= \int _{2}^{x} {dt}/{\log t}}$ . Finally, given a real number $x\ge e^{e^{e}}$ , we set
where
Let ${\cal B}$ be a subset of $\wp $ whose counting function $B(x):=\#\{p\le x: p\in {\cal B}\}$ is such that, for some real number $\beta>0$ ,
so that in particular, for some constant $\beta _{2}$ ,
Any such set ${\cal B}$ satisfying (2.2) will be called a B-set.
Observe that given a B-set and its associated function ${\omega _{\cal B} (n):= \sum _{p\mid n,\, p\in {\cal B}} 1}$ , it is immediate that
and one can show that it follows from the Turán–Kubilius inequality that
implying that the average value and the normal order of the function $\omega _{\cal B}(n)$ are each $\beta \log \log n$ . Moreover, one can establish that, for some positive constant $\beta _{3}$ ,
where ${\Phi (y):= (1/\!{\sqrt {2\pi }}) \int _{-\infty }^{y} e^{-t^{2}/2}\,dt}$ .
Now, given positive real numbers $u<v$ , let us set
Also, given an integer $n>1$ and a prime divisor p of n which is smaller than $P(n)$ , the largest prime divisor of n, we set
so that
Now, consider the arithmetic function
The technique we used in [Reference De Koninck and Kátai3] to prove (1.1) can also be used to determine the mean value of the function $U_{\lambda ,{\cal B}} (n)$ and in fact the following more accurate result.
Theorem 2.1. Given $\lambda \in (0,1]$ and a set of primes ${\cal B}$ satisfying (2.2),
and, for an arbitrarily small $\varepsilon>0$ ,
Estimates (1.1) and (1.2) can be modified to hold for shifted primes. Indeed, we also have the following result.
Theorem 2.2. Given $\lambda \in (0,1]$ , a set of primes ${\cal B}$ satisfying (2.2) and a fixed integer $a\neq 0$ ,
and, moreover, for any arbitrarily small $\varepsilon>0$ ,
3 An extension of Theorem 2.1
Given $\lambda \in (0,1]$ and a set of primes ${\cal B}$ satisfying (2.2), we introduce the arithmetic function $\widetilde {U}_{\lambda ,{\cal B}} (n)$ which counts the number of prime divisors p of n which belong to ${\cal B}$ and for which the next prime divisor q of n also belonging to ${\cal B}$ satisfies ${{\log p/\log q<\lambda }}$ . In short, setting ${ Q_{\cal B}(u,v) := \prod _{u<p<v,\, p\in {\cal B}} p}$ , we can write
In the case of the function $\widetilde {U}_{\lambda ,{\cal B}} (n)$ , we can obtain an asymptotic formula for its average value with greater accuracy than the one obtained for the function $U_{\lambda ,{\cal B}} (n)$ (whose average value was revealed through the estimate (2.4)). Indeed, we can prove the following result.
Theorem 3.1. Let $\lambda \in (0,1]$ and ${\cal B}$ be a set of primes satisfying (2.2). Then
Moreover, for any arbitrarily small $\varepsilon>0$ ,
Before we begin the proof of Theorem 3.1, let us recall a lemma from our recent paper [Reference De Koninck and Kátai4].
Lemma 3.2 [Reference De Koninck and Kátai4, Lemma A].
Let $x\ge e^{e^{100}}$ and let $Y_{1}=Y_{1}(x)$ and $Y_{2}=Y_{2}(x)$ be the functions defined in (2.1). Let $ \pi _{1} < \pi _{2}< \cdots < \pi _{s}$ be s primes located in the interval $(Y_{1},Y_{2})$ . Write their product as $R=\pi _{1}\pi _{2}\cdots \pi _{s}$ . Further, set
and
Assume that $\eta \le K$ , where K is an arbitrary number, and let h be a positive integer satisfying $h\ge 3e^{2} K$ . Then, letting $\phi $ stand for the Euler totient function,
so that in particular, choosing $h=\lfloor \log \log \log x \rfloor $ , there exists a positive constant c such that
Remark 3.3. Observe that, as detailed in [Reference De Koninck and Kátai3], the choice of h in Lemma 3.2 is optimal as it ensures that each of the two terms $x(3e)^{-h}$ and $2Y_{2}^{h}$ is much less than $x/(\log \log x)^{2}$ .
We are now ready for the proof of Theorem 3.1.
Proof of Theorem 3.1.
Let $(p,q)\in {\cal B} \times {\cal B}$ , with $p<q$ , be a pair of prime divisors of an integer n which satisfy
Let us first assume that $Y_{1}<p<q<Y_{2}$ , where $Y_{1}=Y_{1}(x)$ and $Y_{2}=Y_{2}(x)$ are the functions defined in (2.1). Using Lemma 3.2 and estimate (2.3),
First assume that $\beta <1$ . In this case, fixing q and summing over p,
say. On the one hand, recalling that $\beta>0$ (see (2.2)),
On the other hand,
Gathering estimates (3.5) and (3.6) in (3.4), we obtain, in the case $\beta <1$ ,
Finally, it is immediate that (3.7) also holds in the case $\beta =1$ .
In light of estimate (3.7), the sum on the right-hand side of (3.3) can therefore be replaced by
where we could ignore the contribution of the last error term on the right-hand side of (3.7) because
Then, in light of (3.8) and with the help of (2.3), it is clear that the estimate (3.3) can be replaced by
thus completing the proof of (3.1).
The proof of (3.2) rests on the evaluation of $\displaystyle { \widetilde {U}_{\lambda ,{\cal B}}(n)^{2} }$ , which can be handled using an approach similar to that used to obtain (3.1). We will therefore omit this proof.
4 Two B-sets involving congruence classes
Given an integer $k\ge 3$ , let $\ell _{1},\ldots ,\ell _{r}$ be the reduced residue system modulo k, with $r=\phi (k)$ . Then it is clear that each of the residue classes
satisfies condition (2.2) and is therefore a B-set.
A second interesting B-set involves the sum-of-digits function $s_{q}(n)$ which stands for the sum of the base q digits of an integer n (here, $q\ge 2$ ). First note that it is known (see Mauduit and Rivat [Reference Mauduit and Rivat11] as well as Drmota et al. [Reference Drmota, Mauduit and Rivat5]) that if $(k,q-1)=1$ , then there exists a real number $\sigma =\sigma _{q,k}>0$ such that
Clearly, (4.1) guarantees that if, for a given integer $k\ge 3$ , we set
then ${\cal B}_{\ell }$ is indeed a B-set for each $\ell =0,1,\ldots ,k-1$ .
5 Two B-sets involving fractional parts
We first identify a B-set involving fractional parts of prime powers. Let $\alpha ,\sigma \in (0,1)$ . In what follows, $\{y\}$ stands for the fractional part of y. Using the method of exponential sums initiated in 1940 by Vinogradov [Reference Vinogradov13], one can prove that
(see the recent paper of Shubin [Reference Shubin12] for a thorough discussion), where $R(x)\ll x^{\theta (\alpha )+\varepsilon }$ with
In light of (5.1), it is clear that indeed, given $\alpha ,\sigma \in (0,1)$ , the set
is a B-set.
Another B-set involving fractional parts is as follows. Let $\alpha $ be an irrational number. As was shown in 1940 by Vinogradov [Reference Vinogradov13],
Consequently, the sequence $(\alpha p)_{p\in \wp }$ is uniformly distributed modulo 1. Thus,
Our next theorem will depend on the following result of Harman.
Theorem 5.1 [Reference Harman10, Theorem 2.2].
Let $\alpha \in {\mathbb R}$ and suppose that, for integers $a,q$ with $(a,q)=1$ , the inequality
holds. Let $\delta \in (0,1]$ be given. Write
Then, for any real $\beta $ and positive integers $N, H$ such that $N^{\varepsilon } \ll q \ll N^{1-\varepsilon }$ for some $\varepsilon>0$ , we have
(Here $\Lambda (n)$ stands for the von Mangoldt function. Also $\|\theta \|:=\min _{m\in {\mathbb Z}} |\theta -m|$ .)
From Theorem 5.1, one can easily deduce the following result.
Theorem 5.2. Let $\varepsilon>0$ be an arbitrarily small number. Let $\alpha \in (0,1)$ be an irrational number such that, for every real $x>x_{0}(\alpha )$ , there exists a positive integer $q\in [x^{\varepsilon },x^{1-\varepsilon }]$ for which $\|q\alpha \|<1/q$ . Then
Remark 5.3. Observe that the condition imposed on $\alpha $ in Theorem 5.2 is known to hold for almost all real numbers $\alpha $ and for every irrational algebraic number $\alpha $ .
Given $0\le u < v <1$ , let $\alpha $ be an irrational number satisfying the conditions of Theorem 5.2 and set ${\cal B}:=\{p\in \wp : \{\alpha p\}\in [u,v)\}$ . Then it follows from Theorem 5.2 that ${\cal B}$ is a B-set.
6 Disjoint classification of primes
Given an integer $k\ge 2$ , we will now be interested in sets of prime numbers $\wp _{1},\wp _{2},\ldots ,\wp _{k}$ such that
where ${\cal D}$ is a finite (perhaps empty) set of primes. We assume that, for each ${j\in \{1,2,\ldots ,k\}}$ ,
where each $\beta _{j}\in (0,1]$ and $\sum _{j=1}^{k}\beta _{j}=1$ . Such a collection of subsets of primes is called a disjoint classification of primes, a notion we introduced in [Reference De Koninck and Kátai2] a decade ago in order to create large families of normal numbers. Here, we use this concept for a different purpose.
For each $j=1,2,\ldots ,k$ , let $E_{j}$ be a subinterval of $[0,1]$ . Given a large integer n, we will now count those k-tuples of primes $(q_{1},q_{2},\ldots ,q_{k})$ which satisfy the conditions
In fact, let $E(n)$ be the number of these primes $q_{1}$ , that is, primes $q_{1}$ for which there exist primes $q_{2},\ldots ,q_{k}$ satisfying the conditions listed in (6.1).
By using our usual technique and writing $\lambda (I)$ for the length of the interval I, we can prove that, if we set
then,
and moreover that, given any small number $\varepsilon>0$ ,
An analogous result for shifted primes can also be proved, namely that, for any fixed integer $a\neq 0$ ,
7 The sequence $n^{2}+1$
It is known that if $p\mid n^{2}+1$ , then either $p=2$ or $p\equiv 1 \ (\mathrm {mod}\ 4)$ (since $({-1}/p)=1$ if and only if $p\equiv 1 \ (\mathrm {mod}\ 4)$ ). On the one hand,
For the sake of completeness, this can be proved by first observing that
where the last term accounts for the contribution of those primes $p\in (x,x^{2}+1]$ . Since for each prime $p\equiv 1 \ (\mathrm {mod}\ 4)$ , the congruence $n^{2}+1\equiv 0\ (\mathrm {mod}\ p) $ has exactly two solutions, the number of positive integers $n\le x$ for which $n^{2}+1\equiv 0 \ (\mathrm {mod}\ p)$ is equal to $2x/p+O(1)$ . Hence,
which, combined with (7.2), proves (7.1).
On the other hand, using estimate (7.1), it follows from the Turán–Kubilius inequality that
In light of (7.1) and (7.3), we may conclude that both the average value and the normal order of the function $\omega (n^{2}+1)$ are $\log \log n$ .
For convenience, we will now be using the notation
Given a small number $\varepsilon>0$ and $\lambda \in (\varepsilon ,1)$ , let us count those pairs of primes $p<q$ such that
For each such pair of primes $p,q$ , let $E_{p,q}$ be the number of those positive integers $n\le x$ for which conditions (7.4) hold. Using Lemma 3.2, we may write
where the prime ( $^{\prime }$ ) on the above sums indicates that the summation runs over those pairs of primes $p,q$ which satisfy the conditions listed in (7.4).
It is easily established that
Using (7.6) in (7.5), we obtain
Denote by $K_{\varepsilon ,\lambda }(n)$ the number of those primes p for which there exists a prime $q>p$ satisfying the list of conditions (7.4). Since the contribution of those primes $p<Y_{1}$ and $q>Y_{2}$ to the above sums is relatively small, it follows from (7.7) that
Since $\varepsilon $ can be chosen arbitrarily small, it follows from (7.8) that
Moreover, one can show that, given any arbitrarily small number $\delta>0$ ,
We can also prove an analogue of [Reference De Koninck and Kátai3, Theorem 5]. Indeed, let $\delta _{0},\delta _{1},\ldots ,\delta _{k-1} \in (0,1)$ and set $H:=\delta _{0}\delta _{1}\cdots \delta _{k-1}$ . Further, let ${\cal F}$ be the set of all $(k+1)$ -tuples of primes $(p_{0},p_{1},\ldots ,p_{k})$ where $p_{0}<p_{1}<\cdots <p_{k}$ satisfy
Also let $V_{\delta _{0},\delta _{1},\ldots ,\delta _{k-1}}(n)$ stand for the number of those prime divisors $p_{0}$ of $n^{2}+1$ for which
Then we have the following result.
Theorem 7.1. Let $\delta _{0},\delta _{1},\ldots ,\delta _{k-1}$ , H and ${\cal F}$ be as above. Then, as $x\to \infty $ ,
Remark 7.2. It follows from Theorem 7.1 that
On the other hand, most likely, for some positive constant c, the function
is distributed according to the normal law. However, in order to prove this, one would need to compute the asymptotic value of the sum $\sum _{n\le x} V_{\delta _{0},\delta _{1},\ldots ,\delta _{k-1}}(n)^{k}$ with a good remainder term, for every k. We were not successful in doing so.
Acknowledgement
The authors would like to thank the referee for very constructive remarks which helped to improve the quality of this paper.