Primes in Beatty sequence

TL;DR

This paper investigates primes within Beatty sequences generated by irrational slopes, providing bounds on the least such prime, asymptotic counts for primes of specific forms, and their distribution properties.

Contribution

It establishes new upper bounds and asymptotic formulas for primes in Beatty sequences, including primes of polynomial forms and with specified congruence conditions.

Findings

Bound on the least prime in Beatty sequence where polynomial takes prime values

Asymptotic formula for the count of primes of the form loorlpha n+eta

Distribution results for primes in Beatty sequences with congruence restrictions

Abstract

For a polynomial $g(x)$ of deg $k \geq 2$ with integer coefficients and positive integer leading coefficient, we prove an upper bound for the least prime $p$ such that $g(p)$ is in non-homogeneous Beatty sequence $\lbrace \lfloor \alpha n+\beta\rfloor : n=1,2,3, \dots \rbrace$, where $\alpha, \beta \in \mathbb{R}$ with $\alpha >1$ is irrational and we prove an asymptotic formula for the number of primes $p$ such that $g(p)=\lfloor \alpha n+\beta \rfloor.$ Next we obtain an asymptotic formula for number of primes $p$ of the form $p=\lfloor \alpha n+\beta \rfloor$ which also satisfies $p \equiv f \pmod d$ where $f, d$ are integers with $1\leq f < d$ and $(f,d)=1$.

Equations

$$\mathcal{B}(\alpha ,\beta )=\{\lfloor n\alpha +\beta \rfloor :n\in \mathbb{N}\},$$

$$t=\sup \{\rho \in \mathbb{R}:\underset{n\to \infty }{\mathrm{liminf}}{n}^{\rho }\Vert n\gamma \Vert =0\},$$

$$p\le L^{35-16\epsilon }{\alpha }^{2(1-\epsilon )}B{p}_{m+l}^{1+\epsilon },$$

$$p_m\le L^{16}{\alpha }^2<p_{m+1}.$$

$$\displaystyle\sum_{\begin{subarray}{c}p\leq N\\ g(p)\in\mathcal{B}(\alpha,\beta)\end{subarray}}\log p=\frac{1}{\alpha}\sum_{p\leq N}\log p+O\bigg{(}N^{\varepsilon}(Nq^{-\gamma}+N^{1-\gamma/2}+q^{\frac{\gamma}{1-\gamma}}N^{\frac{1-(k+1)\gamma}{1-\gamma}}+q^{\gamma}N^{1-k\gamma})\bigg{)}.$$

$$\sum _{\begin{array}{c}p\le N\\ g(p)\in \mathcal{B}(\alpha ,\beta )\end{array}}\log p=\frac{1}{\alpha }\sum _{p\le N}\log p+O(N^{1-\frac{k\gamma }{t+1}+\epsilon }+N^{1-\frac{\gamma }{2}+\epsilon })$$

$$p\le {\alpha }^{\frac{2k-1}{k\gamma }-\frac{(\gamma +1)\epsilon }{\gamma }}{B}^{\frac{k-1}{k}-\frac{\epsilon }{\gamma }}{p}_{m+l}^{\frac{1}{k}+\frac{\epsilon }{\gamma }},$$

$$p_m\le {\alpha }^{\frac{1+\gamma }{\gamma }}B<p_{m+1}.$$

$$\sum _{\begin{array}{c}n\le x,n\in {\mathcal{B}}_{\alpha ,\beta }\\ n\equiv c(\mathrm{mod}d)\end{array}}\Lambda (n)=\frac{1}{\alpha }\sum _{\begin{array}{c}n\le x\\ n\equiv c(\mathrm{mod}d)\end{array}}\Lambda (n)+O(x^{1-\frac{1}{4\tau +2}+\epsilon }).$$

$$\displaystyle\sum_{\begin{subarray}{c}p\leq N\\ p\in\mathcal{B}(\alpha,\beta)\\ p\equiv f(d)\end{subarray}}\log p=\frac{1}{\alpha}\sum_{\begin{subarray}{c}p\leq N\\ p\equiv f(d)\end{subarray}}\log p+O\bigg{(}N^{\varepsilon}\bigg{(}\frac{N}{q^{1/2}}+N^{1/2}q^{1/2}+N^{3/4}d^{1/2}+\frac{N^{4/5}}{d^{1/5}}\bigg{)}\bigg{)}.$$

$$\sum _{\begin{array}{c}p\le N\\ p\in \mathcal{B}(\alpha ,\beta )\\ p\equiv f(d)\end{array}}\log p=\frac{1}{\alpha }\sum _{\begin{array}{c}p\le N\\ p\equiv f(d)\end{array}}\log p+O(N^{1-\frac{1}{2(1+t)}+\epsilon }+N^{\frac{3}{4}+\epsilon }{d}^{1/2}+N^{\frac{4}{5}+\epsilon }{d}^{-1/5}).$$

$$S(\vartheta)=\sum_{|l|\leq L}\bigg{|}\sum_{\begin{subarray}{c}n\leq N\\ n\equiv f(d)\end{subarray}}\Lambda(n)e(ln\vartheta)\bigg{|},$$

$$p\le {\alpha }^{3-7\epsilon }{B}^{\frac{1}{2}(1-3\epsilon )}{d}^{3-10\epsilon }{p}_{m+l}^{1+3\epsilon }$$

$$p_m\le {\alpha }^{7/3}{B}^{1/2}{d}^{10/3}<p_{m+1}.$$

$$D(M)=\sup_{I\subset[0,1)}\bigg{|}\frac{V(I,M)}{M}-|I|\bigg{|},$$

$$\bigg{\|}\frac{m}{\alpha}+\frac{1-2\beta}{2\alpha}\bigg{\|}<\frac{1}{2\alpha}.$$

$$\#\{m\leq N:m\in\mathcal{B}(\alpha,\beta)\}=\sum_{m\leq N}\chi_{\frac{1}{2\alpha}}\bigg{(}\frac{m}{\alpha}+\frac{1-2\beta}{2\alpha}\bigg{)}$$

$${\chi }_{\delta }(\theta )=\{\begin{array}{ll}1& \text{if }\Vert \theta \Vert <\delta ,\\ 0& \text{otherwise}\end{array}$$

$$\#\{p\leq N:g(p)\in\mathcal{B}(\alpha,\beta)\}=\sum_{p\leq N}\chi_{\frac{1}{2\alpha}}\bigg{(}\frac{g(p)}{\alpha}+\frac{1-2\beta}{2\alpha}\bigg{)}.$$

$$2\delta -\frac{1}{L+1}+\sum _{1\le l\le L}{C}_l^{-}e(l\theta )\le {\chi }_{\delta }(\theta )\le 2\delta +\frac{1}{L+1}+\sum _{1\le l\le L}{C}_l^{+}e(l\theta ),$$

$$\displaystyle\sum_{n\leq N}\Lambda(n)\chi_{\frac{1}{2\alpha}}\bigg{(}\frac{g(n)}{\alpha}+\frac{1-2\beta}{2\alpha}\bigg{)}=$$

$$\displaystyle+O\bigg{(}\sum_{1\leq|l|\leq L}|C_{l}|\bigg{|}\sum_{n\leq N}\Lambda(n)e\bigg{(}\frac{lg(n)}{\alpha}\bigg{)}\bigg{|}\bigg{)},$$

$$|q\alpha -a|<q^{-1}.$$

$$\sum_{l\leq L}\bigg{|}\sum_{n\leq N}\Lambda(n)e(lf(n))\bigg{|}\ll(NL)^{1+\varepsilon}(q^{-1}+N^{-1/2}+qN^{-k}L^{-1})^{\gamma}.$$

$$\bigg{\|}\frac{p}{\alpha}+\frac{1-2\beta}{2\alpha}\bigg{\|}<\frac{1}{2\alpha}.$$

$$\displaystyle\sum_{\begin{subarray}{c}n\leq N\\ n\equiv f(d)\end{subarray}}\Lambda(n)\chi_{\frac{1}{2\alpha}}\bigg{(}\frac{n}{\alpha}+\frac{1-2\beta}{2\alpha}\bigg{)}=$$

$$\displaystyle+O\bigg{(}\sum_{1\leq|l|\leq L}|C_{l}|\sum_{\begin{subarray}{c}n\leq N\\ n\equiv f(d)\end{subarray}}\Lambda(n)e(ln/\alpha)\bigg{)},$$

$$S(\vartheta)\ll_{\varepsilon}(NL)^{\varepsilon}\bigg{(}\frac{NL}{q^{1/2}}+L^{1/2}N^{1/2}q^{1/2}+LN^{3/4}d^{1/2}+\frac{LN^{4/5}}{d^{1/5}}\bigg{)}.$$

$$\displaystyle\sum_{1\leq|l|\leq L}C_{l}\sum_{n\leq N}\Lambda(n)e\bigg{(}l\bigg{(}\frac{g(n)}{\alpha}+\frac{1-2\beta}{2\alpha}\bigg{)}\bigg{)}\ll_{\varepsilon}$$

$$+N^{1+\epsilon }{q}^{\gamma }{N}^{-k\gamma }.$$

$$\displaystyle\sum_{n\leq N}\Lambda(n)\chi_{\frac{1}{2\alpha}}\bigg{(}\frac{g(n)}{\alpha}+\frac{1-2\beta}{2\alpha}\bigg{)}=\bigg{(}\frac{1}{\alpha}\bigg{)}\sum_{n\leq N}\Lambda(n)+O\bigg{(}\frac{N}{L+1}\bigg{)}$$

$$\displaystyle+O\bigg{(}N^{1+\varepsilon}L^{\varepsilon}(q^{-\gamma}+N^{-\gamma/2}+q^{\gamma}N^{-k\gamma}L^{-\gamma})+N^{1+\varepsilon}q^{\gamma}N^{-k\gamma}\bigg{)}.$$

Full text

1. Introduction

Given a real number $\alpha>0$ and a non-negative real $\beta$, the Beatty sequence associated with $\alpha,\beta$ is defined by

[TABLE]

where $\lfloor x\rfloor$ denotes the largest integer less than or equal to $x$. If $\alpha$ is rational, then $\mathcal{B}(\alpha,\beta)$ is union of residue classes, hence we always assume that $\alpha$ is irrational. An irrational number $\gamma$ is said to be of finite type $t\geq 1$ if

[TABLE]

where $\|x\|$ is the distance of $x$ from nearest integer. In 2016, Jörn Steuding and Marc Technau [8] proved that, for every $\varepsilon>0$ there exists a computable positive integer $l$ such that for every irrational $\alpha>1$ the least prime $p$ in the Beatty sequence $\mathcal{B}(\alpha,\beta)$ is satisfies the inequality

[TABLE]

where $B=\max\{1,\beta\}$, $L=\log(2\alpha B),$ $p_{n}$ denotes the numerator of the $n^{th}$ convergent to the regular continued fraction expansion of $\alpha=[a_{0},a_{1},\dots]$ and m is the unique integer such that

[TABLE]

The first result in this paper is the following

Theorem 1**.**

Let $g(x)=a_{0}+a_{1}x+\dots+a_{k}x^{k},$ where $a_{0},\dots,a_{k}\in\mathbb{Z}$ with $a_{k}\geq 1$ and $k\geq 2$. Put $\gamma=4^{1-k}.$ Then for any positive integer $N\geq 3,$ positive real number $\alpha$ with \bigg{|}\frac{a_{k}}{\alpha}-\frac{a}{q}\bigg{|}\leq\frac{1}{q^{2}}, $(a,q)=1$ and any $\varepsilon>0$ we have

[TABLE]

In particular, if $a_{k}/\alpha$ is an irrational number of finite type $t>0$ then we have

[TABLE]

Theorem 2**.**

Let $g(x)=a_{0}+a_{1}x+\dots+a_{k}x^{k},$ where $a_{0},\dots,a_{k}\in\mathbb{Z}$ with $a_{k}\geq 1$ and $k\geq 2$. Put $\gamma=4^{1-k}.$ For every $\varepsilon>0,$ there exits a computable positive integer l such that for every irrational $\alpha>1$ the least prime number $p$ such that $g(p)$ is contained in the Beatty sequence $\mathcal{B}(\alpha,\beta)$ satisfies the inequality,

[TABLE]

where $B=\max\{1,\beta\}$, $p_{n}$ denotes the numerator of the $n^{th}$ convergent to the regular continued fraction expansion of $\frac{\alpha}{a_{k}}$ and m is the unique integer such that

[TABLE]

For irrational $\alpha$ of finite type $\tau=\tau(\alpha)$, Banks and Yeager proved in ([2], Theorem 2) that for any fixed $\varepsilon>0,$ for all integers $1\leq c<d<N^{\frac{1}{4\tau+2}}$ with gcd (c, d)=1, we have

[TABLE]

The following theorem improves the error term.

Theorem 3**.**

For any positive integers $N\geq 3$, $1\leq f<d\leq\min\{q^{1/2},N^{1/6}\}$ such that $(f,d)=1$ and positive real number $\alpha$ with \bigg{|}\frac{1}{\alpha}-\frac{a}{q}\bigg{|}\leq\frac{1}{q^{2}} with $(a,q)=1,$ for any $\varepsilon>0$, we have

[TABLE]

Furthermore if $\frac{1}{\alpha}$ is an irrational number of finite type $t>0$ then for all integers $1\leq f<d\leq\min\{N^{\frac{1}{2(t+1)}},N^{1/6}\}$ with $(f,d)=1$ and for any $0<\varepsilon<\frac{1}{4(t+1)},$ we have

[TABLE]

The proof of Theorem 3 depends on estimation of exponential sum of the type

[TABLE]

where $\vartheta$ is irrational, $L,N\geq 1$ and $f<d,(f,d)=1.$

We obtain an upper bound for $S(\vartheta)$ in Proposition 2 which is of independent interest.

Remark 1**.**

Let $d,f$ be natural numbers such that $1\leq f<d\leq 500$ and $(f,d)=1$. For every $\varepsilon>0$ there exists a computable positive integer $l$ such that for every irrational $\alpha>1;$ the least prime number $p\in\mathcal{B}(\alpha,\beta)$ such that $p\equiv f\pmod{d}$ satisfies the inequality,

[TABLE]

where $B=$max$\{1,\beta\}$ and $p_{n}$ denotes the numerator of the $n^{th}$ convergent to the regular continued fraction expansion of $\alpha$ and m is the unique integer such that,

[TABLE]

This fact can be proved in a similar way as Theorem 2 using Corollary 1.6 of [3].

Acknowledgement. I would like to express my sincere thanks to my thesis supervisor Anirban Mukhopadhyay for his valuable and constructive suggestions during the planning and development of this paper. I would also like to thank Marc Technau for suggesting important changes in an earlier version of this manuscripts.

2. Notation

Throughout this paper, the implied constants in the symbols $O$ and $\ll$ may depend on $\alpha$ and $\varepsilon$ otherwise are absolute. We recall that the notation $f=O(g)$ and $f\ll g$ are equivalent to the assertion that the inequality $|f|\leq cg$ holds for some constant $c>0$. The notation $f\approx g$ means that $f\ll g$ and $f\gg g$. It is important to note that our bounds are uniform with respect to all of the involved parameters other than $\alpha,\varepsilon$ and degree of the polynomial $k$; in particular, our bounds are uniform with respect to $\beta$.

The letters $a,d,f,q$ always denote non-negative integers and $m,n,l,u,v$ and $t$ denotes integers. We use $\lfloor x\rfloor$ and $\{x\}$ to denote the greatest integer less than or equal to $x$ and the fractional part of $x$ respectively. Finally, recall that the discrepancy $D(M)$ of a sequence of (not necessarily distinct) real numbers $a_{1},a_{2}\dots,a_{M}\in[0,1)$ is defined by

[TABLE]

where the supremum is taken over all sub-intervals $I$ of $[0,1)$, $V(I,M)$ is the number of positive integers $m\leq M$ such that $a_{m}\in I$ and $|I|$ is the length of $I$.

3. Preliminaries

3.1. Case of polynomial values of prime

Note that an integer $m\in\mathcal{B}(\alpha,\beta)$ if and only if

\frac{m}{\alpha}\in\bigg{(}\frac{\beta-1}{\alpha},\frac{\beta}{\alpha}\bigg{]}\pmod{1} and $m>\alpha+\beta-1.$ This is equivalent to

[TABLE]

Hence

[TABLE]

where $\chi_{\delta}$ for $\delta>0$ is defined by,

[TABLE]

for $\theta\in\mathbb{R}$. Let $g(x)=a_{0}+a_{1}x+\dots+a_{k}x^{k},$ where $a_{0},\dots,a_{k}\in\mathbb{Z}$, $a_{k}\geq 1$. Therefore

[TABLE]

Lemma 1**.**

([5], Lemma 2.1). For any $L\in\mathbb{N}$ there are coefficients $C^{\pm}_{l}$ such that

[TABLE]

with |C_{l}^{\pm}|\leq\min\bigg{(}2\delta+\frac{1}{L+1},\frac{3}{2l}\bigg{)}.

Using Lemma 1 we get

[TABLE]

where |C_{l}|\leq\min\bigg{(}\frac{1}{\alpha}+\frac{1}{L+1},\frac{3}{2l}\bigg{)}. To estimate the exponential sum we use the following Proposition

Proposition 1**.**

(Equation (22), [4]) Suppose $\varepsilon>0$ is given. Let $f(x)$ be a real valued polynomial in $x$ of degree $k\geq 2.$ Put $\gamma=4^{1-k}.$ Suppose $\alpha$ is the leading coefficient of $f$ and there are integers $a,q$ with $(a,q)=1$ such that

[TABLE]

Then we have

[TABLE]

3.2. Case of primes in arithmetic progression

Now we are interested in prime numbers p of the form $p\equiv f$ mod d, which is in $\mathcal{B}(\alpha,\beta)$, where $(f,d)=1$ and $f<d$. As we discussed above, in order to find a prime number $p\in\mathcal{B}(\alpha,\beta)$ and we need to show that

[TABLE]

By Lemma 1, we have

[TABLE]

where |C_{l}|\leq\min\bigg{(}\frac{1}{\alpha}+\frac{1}{L+1},\frac{3}{2l}\bigg{)}. Now we want to estimate the exponential sum of the form (2). To estimate the exponential sum we use the following Proposition

Proposition 2**.**

Let $S(\vartheta)$ is defined by 2 with \bigg{|}\vartheta-\frac{a}{q}\bigg{|}\leq q^{-2}, where $a$ and $q$ are positive integers satisfying (a,q)=1. Then for any real number $\varepsilon>0;$ we have

[TABLE]

We will give the proof of Proposition 2 in Section 6.

4. Proof of Theorem 1 and Theorem 2

In the previous section we stated essential results to prove Theorem 1 and Theorem 2. In this section we will give proof of these theorems. Proof of Theorem 1: It follows from Proposition 1 and partial summation formula

[TABLE]

By (3.1) and (4) we have

[TABLE]

Choosing $L=q^{\frac{-\gamma}{1-\gamma}}N^{\frac{k\gamma}{1-\gamma}}$ we have

[TABLE]

This leads to

[TABLE]

where

[TABLE]

The number of prime powers $p^{\nu}\leq N$ with $\nu\geq 2$ is $O(\pi(N^{1/2})),$ thus we have

[TABLE]

Suppose we assume $\frac{a_{k}}{\alpha}$ is an irrational number of finite type $t.$ Using Dirichlet’s approximation theorem with $Q=N^{\frac{kt}{t+1}}$, we obtain a rational $p/q$ with $1\leq q\leq N^{\frac{kt}{t+1}}$ such that

[TABLE]

By definition of finite type of irrational, for any positive $\varepsilon,$ there is positive constant $c$ such that

[TABLE]

Then by (10) and (11) there exists a convergent to the simple continued fraction expansion of $\frac{a_{k}}{\alpha}$ whose denominator satisfies

[TABLE]

Therefore by (9) and (12) we obtain

[TABLE]

This completes the proof of the Theorem 1.

Proof of Theorem 2: By (7) and (8) we have

[TABLE]

where

[TABLE]

By Lemma 2, the second sum on the left hand side of (13) is $<1.04~{}(\alpha+\beta-1).$

Therefore, we have

[TABLE]

Notice that the last term is negative, it is obviously bounded by

[TABLE]

We will use inequality $(2.18)$ Rosser and Schoenfeld [7] for $\pi(x)$, we have

[TABLE]

we also use inequality (3.16) of Rosser and Schoenfeld which is,

[TABLE]

for $N\geq 41$. Therefore we obtain

[TABLE]

We thus find a prime $p\leq N$ and $p^{2}\in\mathcal{B}(\alpha,\beta)$ if we show that the following inequality

[TABLE]

which we may also replace by

[TABLE]

By (14) we have,

[TABLE]

and appropriate absolute constant $C(\varepsilon)$ depending only on $\varepsilon$ but not an $\alpha$.

Obviously $N$ need to be larger than Max$\{\alpha^{2/\gamma},B\}$ and q larger than $\alpha^{1/\gamma}$. We shall take both N and q somewhat larger so that above inequality holds, now choose

[TABLE]

with some large parameter $\eta$ to be specified later and B=max$\{1,\beta\}$. Then the latter inequality can be rewritten as

[TABLE]

Since $k\geq 2$ and $\gamma=4^{1-k}$ assuming $\varepsilon<\frac{\gamma^{2}}{2(2k+\gamma)}$, as we may, all exponents of $\alpha,B$ and $\eta$ are negative. Therefore the above inequality is satisfied for all sufficiently large $\eta$, say $\eta\geq\eta_{0}.$ Since $\eta$ is interwined with $q$ a little care needs to be taken. In order to find a suitable $\eta$ recall $\alpha$ is irrational. Hence, by Dirichlet’s approximation theorem, there are infinitely many solution $\frac{a}{q}$ to inequality $|\alpha-\frac{a}{q}|<\frac{1}{q^{2}};$ in view of $\mathbf{a}=\frac{1}{\alpha}$ we may take the reciprocals of the convergents $\frac{p_{n}}{q_{n}}$ to the continued fraction expansion of $\alpha.$ We shall choose $l$ such that $\eta_{0}\leq\frac{p_{m+l}}{p_{m}},$ where $m$ is defined by (1), for then the choice $q=p_{m+l}$ will yield an $\eta\geq\eta_{0}.$ The choice of $\eta$ follows from $(12)$ of [8]. Therefore the choice of $l$ as it depends on $\eta.$ This completes the proof of the Theorem 2.

5. Proof of theorem 3

The present section is devoted to a proof of Theorem 3. Proof of Theorem 3: It follows from Proposition 2 and partial summation formula

[TABLE]

By (3.2) and (5), we obtain

[TABLE]

Choose

[TABLE]

Therefore we obtain an estimate

[TABLE]

We rewrite above equality as

[TABLE]

By $(4.3.3)$ of [6] we have

[TABLE]

Thus we have

[TABLE]

Suppose we assume $\frac{1}{\alpha}$ is an irrational number of finite type $t$. By using Dirichlet’s approximation theorem with $Q=N^{\frac{t}{1+t}}$, we obtain a rational $p/q$ with $1\leq q\leq N^{\frac{t}{1+t}}$ such that

[TABLE]

And by definition of finite type of irrational, for any positive $\varepsilon,$ there is positive constant $c$ such that

[TABLE]

Then by (16) and (17) there exists a convergent to the simple continued fraction expansion of $\frac{1}{\alpha}$ whose denominator satisfies

[TABLE]

Therefore by (5) and (18) we obtain

[TABLE]

This completes the proof of the Theorem 3.

6. Proof of proposition 2

Proof of Proposition 2 is based on work of Balog and Perelli [1].

6.1. Some Lemmas

Here we list several lemmas required for the proof. The following lemma gives explicit bound for average of von Mangoldt function

Lemma 2**.**

[7]** For any $N\in\mathbb{N}$

[TABLE]

for some constant $c_{0}$, where one may take $c_{0}=1.04.$

Lemma 3**.**

[TABLE]

Lemma 4**.**

[10]** Suppose that X, Y $\geq 1$ are positive integers, Also suppose that $|\alpha-a/q|<q^{-2}$, where $\alpha$ is a real number, $a$ and $q$ integers satisfying $(a,q)=1$. Then

[TABLE]

[TABLE]

Lemma 5**.**

[9]** For any real number $\vartheta$ and natural numbers $N,l$ and $1\leq f<d$ such that $(f,d)=1,$ we have

[TABLE]

where

[TABLE]

and

[TABLE]

Here $U$ is an arbitrary parameters to be chosen later satisfying $1\leq U\leq N^{1/2}.$

Lemma 6**.**

Suppose that $\varepsilon>0$ and that $\phi(u)$ and $\psi(v)$ are real valued functions such that $|\phi(u)|\ll T,$ $|\psi(v)|\ll F.$ Suppose that $|\vartheta-a/q|<q^{-2}$, where $\vartheta$ is a real number, $a$ and $q$ integers satisfying $(a,q)=1$. For positive integers $N,W,X,$ and $L$ write

[TABLE]

Then

[TABLE]

Proof.

For the moment we shall ignore the condition $uv\leq N$ in (19). Consider

[TABLE]

We observe that

[TABLE]

where

[TABLE]

By using Cauchy Schwarz inequality we obtain

[TABLE]

where

[TABLE]

We may write $R_{1}$ in the form

[TABLE]

where

[TABLE]

We apply Lemma 3 for innermost sum of (6.1), we get

[TABLE]

Let $r=lkd$ so that $1\leq|r|\leq 2LXd$ and $r$ will run through all the integers in the interval above, also number of representations of $r$ is not more than $d_{2}(r).$ Therefore we have

[TABLE]

Then by using Lemma 4 we obtain

[TABLE]

By (6.1) and (24) we have

[TABLE]

Thus Lemma follows from (21) and (25) . ∎

Lemma 7**.**

Suppose we have the hypotheses and notations of Lemma 6 with either $\phi(x)=1$ or $\phi(x)=\log x$ for all x. Then

[TABLE]

Proof.

The $\log x$ factor may easily be removed by partial summation formula so we presume that $\phi(x)\equiv 1.$ Again we may ignore the condition $uv\leq N.$ Therefore we need to estimate

[TABLE]

where $\bar{u}$ is defined by $u\bar{u}\equiv 1\pmod{d}.$ Then by using Lemma 3, we have

[TABLE]

Let $r=lvd$ so that $1\leq|r|\leq 2LXd$ and $r$ will run through all the integers in the interval above, also number of representations of $r$ is not more than $d_{2}(r).$ Therefore we have

[TABLE]

Thus (26) follows from (27) and Lemma 4. ∎

Proof of the proposition 2: We may assume that

[TABLE]

otherwise (5) is a consequence of the trivial bound,

[TABLE]

Using Lemma 5 we have the following sums to estimate

[TABLE]

By dyadic division we write

[TABLE]

where

[TABLE]

Then using Lemma 7, we get

[TABLE]

$S^{\prime}_{2}$ can be estimated similarly as $S^{\prime}_{1}$ by partitioning into dyadic subsums say $S_{2t}$. We estimate $S_{2t}$ using Lemma 7, and we get

[TABLE]

We write $S^{\prime}_{3}=S^{\prime}_{31}+S^{\prime}_{32},$ where

[TABLE]

By dividing $S^{\prime}_{31}$ dyadically we obtain

[TABLE]

where

[TABLE]

Then using Lemma 6

[TABLE]

Similarly we can show that $S^{\prime}_{32}$ has the same upper bound. Therefore

[TABLE]

Then (5) follows from (29) with the chioce of

[TABLE]

and the observation $q\leq L^{1/2}N^{1/2}q^{1/2}.$