On products of primes and square-free integers in arithmetic progressions

TL;DR

This paper derives an asymptotic formula for counting representations of residue classes as products of a prime and a square-free integer, advancing understanding of prime-related structures in arithmetic progressions.

Contribution

It provides a new asymptotic formula for representing residue classes as products of primes and square-free integers, addressing a relaxed form of a conjecture by Erdős, Odlyzko, and Sárközy.

Findings

Established an asymptotic count for such representations.

Connected the problem to a relaxed version of a classical conjecture.

Enhanced understanding of prime and square-free integer distributions in residue classes.

Abstract

We obtain an asymptotic formula for the number of ways to represent every reduced residue class as a product of a prime and square-free integer. This may be considered as a relaxed version of a conjecture of Erd\"os, Odlyzko, and S\'ark\"ozy.

Equations

$$p_1{p}_2\equiv a(\mathrm{mod}m)$$

$$\#\left\{(p,s):ps\equiv a(\mathrm{mod}q),p\le P,s\le S,{\mu }^2(s)=1,(ps,q)=1\right\}$$

$${\mu }^2(n)=\sum _{d^2|n}\mu (d),$$

$${\pi }_q(P)$$

$$s_q(S)$$

$$\#\left\{(p,s):ps\equiv a(\mathrm{mod}q),p\le P,s\le S,{\mu }^2(s)=1,(ps,q)=1\right\}.$$

$${\mathcal{N}}_{a,q}^{\#}(P,S)=\frac{{\pi }_q(P)s_q(S)}{q}+O\left((PS{)}^{o(1)}{S}^{1/2}E\right),$$

$$E=\begin{cases}\displaystyle Pq^{-1}&\mbox{if $q\leq(\log P)^{A}$,}\\[2.0pt] \displaystyle\frac{P}{q^{3/4}}+\frac{P^{9/10}}{q^{3/8}}&\mbox{if $(\log P)^{A}<q<P^{3/4}$,}\\[10.0pt] \displaystyle\frac{P^{31/32}}{q^{(1-\varepsilon)/2}}+\frac{P^{5/6}}{q^{(3/4-\varepsilon)/2}}&\mbox{if $P^{3/4}\leq q$.}\end{cases}$$

$$\frac{{\pi }_q(P)s_q(S)}{q}$$

$$\gg P^{1+o(1)}S{q}^{-1}$$

$$S^2\gg (PS{)}^{\epsilon }q\text{and}{P}^4{S}^{20}\gg (PS{)}^{\epsilon }{q}^{25}.$$

$$P{S}^{16}\gg (PS{)}^{\epsilon }{q}^{16}\text{and}{P}^4{S}^{12}\gg (PS{)}^{\epsilon }{q}^{15}.$$

$$S_q(a;x)=\sum _{\begin{array}{c}p\le x\\ (p,q)=1\end{array}}{\mathbf{e}}_q(a\bar{p}).$$

$$S_q(a;x)=O(B_q(x)),$$

$$B_{q}(x)=\begin{cases}\displaystyle x^{1+o(1)}q^{-1}&\mbox{if $q\leq(\log x)^{A}$,}\\[5.0pt] \displaystyle(q^{-1/2}x+q^{1/4}x^{4/5})x^{o(1)}&\mbox{if $(\log x)^{A}<q<x^{3/4}$,}\\[5.0pt] \displaystyle(x^{15/16}+q^{1/4}x^{2/3})q^{\varepsilon}&\mbox{if $x^{3/4}\leq q$.}\\ \end{cases}$$

$${\mathcal{N}}_{a,q}(P,S)=\#\left\{(p,s):ps\equiv a(\mathrm{mod}q),p\le P,s\le S,(ps,q)=1\right\}$$

$${\mathcal{N}}_{a,q}(P,S)\ll \left(\frac{PS}{q}+1\right)(PS{)}^{o(1)}.$$

$$\ll \log (kq)\ll \log (PS+q)\ll \log (PS)\ll (PS{)}^{o(1)},$$

$$N_q(P,S)=\#\{(p,s):p\le P,s\le S,(ps,q)=1\}.$$

$${\mathcal{N}}_{a,q}(P,S)=\frac{N_q(P,S)}{q}+O(B_q(P)),$$

$$s\equiv a\bar{p}(\mathrm{mod}q)$$

$$N_q(P,S)=\frac{\phi (q){\pi }_q(P)S}{q}+O(P^{1+o(1)}).$$

$$\sum_{d|n}\mu(d)=\begin{cases}1&\mbox{if $n=1$,}\\ 0&\mbox{otherwise.}\end{cases}$$

$$N_q(P,S)$$

$$={\pi }_q(P)\sum _{s\le S}\sum _{\begin{array}{c}d|s\\ d|q\end{array}}\mu (d)$$

$$={\pi }_q(P)\left(\frac{\phi (q)S}{q}+O(\tau (q))\right)$$

$$=\frac{\phi (q){\pi }_q(P)S}{q}+O(P^{1+o(1)}).$$

$$s_q(S)=\frac{\phi (q)}{q}\prod _{p\nmid q}\left(1-\frac{1}{p^2}\right)S+O(S^{1/2}{q}^{o(1)}).$$

$$s_q(S)$$

$$=\sum _{\begin{array}{c}d\le S^{1/2}\\ (d,q)=1\end{array}}\mu (d)\sum _{\begin{array}{c}s\le S/d^2\end{array}}\sum _{\begin{array}{c}r|s\\ r|q\end{array}}\mu (r).$$

$$s_q(S)$$

$$=\frac{\phi (q)}{q}\left(\sum _{\begin{array}{c}d=1\\ (d,q)=1\end{array}}^{\infty }\frac{\mu (d)}{d^2}-\sum _{\begin{array}{c}d>S^{1/2}\\ (d,q)=1\end{array}}\frac{\mu (d)}{d^2}\right)S+O(S^{1/2}\tau (q))$$

$$=\frac{\phi (q)}{q}\prod _{p\nmid q}\left(1-\frac{1}{p^2}\right)S+O(S^{1/2}{q}^{o(1)}),$$

$$\phi (q)=q\prod _{p|q}\left(1-\frac{1}{q}\right)=q\sum _{r|q}\frac{\mu (r)}{r}.$$

Full text

1. Introduction

A conjecture of Erdös, Odlyzko, and Sárközy [4] asks if for every reduced residue class $a$ modulo $m$ can be represented as a product

[TABLE]

for two primes $p_{1},p_{2}\leq m$. Friedlander, Kurlberg, and Shparlinski [6] considered an average of (1.1) over $a$ and $m$, and also various modification of (1.1). Garaev [8, 9] improved on these modifications. Other interesting variants of (1.1) had also been considered by Baker [1], Ramaré & Walker [12], Shparlinski [13, 14], Walker [15].

In this paper, we are concerned with bounding the quantity

[TABLE]

for $(a,q)=1$. This may also be viewed as a multiplicative analogue in the setting of finite fields of a result of Estermann [5]. Estermann [5] showed that all sufficiently large positive integer can be written as a sum of a prime and a square-free integer, see also [10, 11]. Recently, Dudek [3] showed that this is true for all positive integer greater than two.

Our method uses the nice factoring property of the characteristic function for square-free integers

[TABLE]

together with bounds for Kloosterman sums over primes supplied by Fourvy & Shparlinski [7], extending those previous result of Garaev [8].

2. Notation

The notation $U=O(V)$ is equivalent to $U\ll V$ and means there exist an absolute constant $C>0$ such that $U\leq CV$. Exclusively $p$ is a prime number, $\mu$ the Möbius function, $\tau(n)$ is the number of positive divisors of $n$, and $\varphi(n)$ is the number of positive integer up to $n$ coprime to $n$.

3. Result

We denote

[TABLE]

to be the number of square-free integers up to $S$ coprime to $q$.

Finally, for $(a,q)=1$, denote ${\mathcal{N}}_{a,q}^{\#}(P,S)$ by the quantity

[TABLE]

Theorem 3.1**.**

For all fixed $A,\varepsilon>0$, we have

[TABLE]

uniformly for $q\leq P^{O(1)}$ and $(a,q)=1$, where

[TABLE]

The main term in Theorem 3.1 is

[TABLE]

since $q\leq P^{O(1)}$. It follows that ${\mathcal{N}}_{a,q}^{\#}(P,S)>0$ when $P\rightarrow\infty$ if either one of the following three conditions below holds.

1. (1)

$q\leq(\log P)^{A}$ and there exist $\varepsilon>0$ such that $S\gg P^{\varepsilon}$. 2. (2)

$(\log P)^{A}<q<P^{3/4}$ and there exist $\varepsilon>0$ such that

[TABLE] 3. (3)

$P^{3/4}\leq q$ and there an $\varepsilon>0$ such that

[TABLE]

4. Preliminaries

For $(a,q)=1$, we denote the Kloosterman sum over primes

[TABLE]

Here $\mathbf{e}_{q}(x)=\exp(2\pi ix/q)$ and $\overline{p}$ is the multiplicative inverse for $p$ modulo $q$. Bounds for $q$ prime had been obtained by Garaev [8]. Fouvry and Shparlinski [7] extended these results for composite $q$. We gather Theorem 3.1, 3.2 and Equation (3.13) from [7] into the following lemma.

Lemma 4.1**.**

For every fixed $A,\varepsilon>0$, we have

[TABLE]

uniformly for integer $q\geq 2$, $(a,q)=1$, and $x\geq 2$. Here

[TABLE]

Denote

[TABLE]

for $(a,q)=1$. Below, we provide an upper bounds for ${\mathcal{N}}_{a,q}(P,S)$.

Lemma 4.2**.**

For $q\leq P^{O(1)}$, we have

[TABLE]

Proof.

We count the number of solution to $ps=a+kq.$ Therefore we bound $k\ll(PS/q+1)$. For each $a+kq$, the number of distinct prime factor is no more than

[TABLE]

from our upper bound on $k$.     \sqcap$$\sqcup

Denote

[TABLE]

We relate the quantity ${\mathcal{N}}_{a,q}(P,S)$ with $N_{q}(P,S)$.

Lemma 4.3**.**

For all fixed $\varepsilon>0$, we have

[TABLE]

uniformly for $(a,q)=1$, where $B_{q}$ is defined as in Lemma 4.1.

Proof.

We interpret this as a uniform distribution problem. Namely we consider

[TABLE]

which fall in the interval $[1,S]$. The result follows from Lemma 4.1 applied with the Erdös-Turán inequality, see [2].     \sqcap$$\sqcup

We provide a bound for $N_{q}(P,S)$.

Lemma 4.4**.**

For $q\leq P^{O(1)}$, we have

[TABLE]

Proof.

Note the identity

[TABLE]

We have

[TABLE]

\sqcap$$\sqcup

We also provide a bound for $s_{q}(S)$.

Lemma 4.5**.**

We have

[TABLE]

Proof.

In a first step

[TABLE]

Interchanging summation and completing the series, we get

[TABLE]

by noting that

[TABLE]

\sqcap$$\sqcup

5. Proof of Theorem 3.1

Using (1.2), we obtain

[TABLE]

where

[TABLE]

Here $D=D(P,S)$ is a parameter that will be chosen later.

By Lemma 4.2, we bound

[TABLE]

By Lemma 4.3 and 4.4 we get

[TABLE]

Completing the series in the summation over $d$, we assert

[TABLE]

where the last line follows from applying Lemma 4.5.

Now we set

[TABLE]

Then the last two terms in (5.1) are equal and it follows

[TABLE]

If $q\leq(\log P)^{A}$ then the error term above is majorised by

[TABLE]

If $(\log P)^{A}<q<P^{3/4}$ then the error term above is majorised by

[TABLE]

Lastly, if $P^{3/4}\leq q$ then the error term above is majorised by

[TABLE]

The result follows.

Acknowledgement

The author thanks I. E. Shparlinski for the problem and helpful comments together with Liangyi Zhao. The author also thanks the referee for helpful comments. This work is supported by an Australian Government Research Training Program (RTP) Scholarship.