Representation of an integer as the sum of a prime in arithmetic progression and a square-free integer

TL;DR

This paper develops an abstract circle method to estimate the number of representations of large integers as the sum of a prime in an arithmetic progression and a square-free integer, uniformly for small moduli.

Contribution

It introduces a novel application of the local model approach to the circle method for problems involving primes and square-free integers.

Findings

Provides an estimate for the weighted count of such representations

Achieves uniformity for small moduli q

Extends circle method techniques to new additive problems

Abstract

Uniformly for small $q$ and $(a,q)=1$, we obtain an estimate for the weighted number of ways a sufficiently large integer can be represented as the sum of a prime congruent to $a$ modulo $q$ and a square-free integer. Our method is based on the notion of local model developed by Ramar\'e and may be viewed as an abstract circle method.

Equations

$$a\mid b$$

$$a\nmid b$$

$$p$$

$$p^{\alpha }\mid \mid n$$

$$(a_1,\dots ,a_n)$$

$$[b_1,\dots ,b_n]$$

$$\sigma (n)$$

$$\phi (n)$$

$$\mu$$

$$\Lambda$$

$$a\equiv b[k]$$

$${\mathbf{e}}_r(t)$$

$$\text{sideset}{}^{\ast }{\sum }_{a=1}^n$$

$$c_r(k)$$

$$\text{mathbbm}{1}_S$$

$$\mathcal{L}$$

$${\mathcal{R}}_{a,q}(N)=\sum _{\begin{array}{c}N=p+n\\ p\equiv a[q]\end{array}}{\mu }^2(n)\log p$$

$${\mathcal{R}}_{a,q}(N)={\mathfrak{S}}_{a,q}(N)N\left\{1+O_{C_1,C_2}\left({\mathcal{L}}^{-C_2}\right)\right\}$$

$${\mathfrak{S}}_{a,q}(N)$$

$$\times \prod _{p^2\mid q}\left(1-\frac{c_p(N-a)+c_{p^2}(N-a)}{p^2-1}\right).$$

$${\mathfrak{S}}_{a,q}(N)$$

$$=\text{mathbbm}{1}_{\begin{array}{c}\forall p,p^2\nmid (q,N-a)\end{array}}\cdot \frac{6}{\phi (q){\pi }^2}\prod _{\begin{array}{c}p\mid N\\ p\nmid q\end{array}}\left(1+\frac{1}{p^2-1}\right)\prod _{\begin{array}{c}p=2\\ p\nmid Nq\end{array}}^{\infty }\left(1-\frac{1}{(p^2-1)(p-1)}\right)$$

$$\times \prod _{\begin{array}{c}p\mid \mid q\\ p\mid (N-a)\end{array}}\left(1-\frac{1}{p+1}\right)\prod _{\begin{array}{c}p\mid \mid q\\ p\nmid (N-a)\end{array}}\left(1+\frac{1}{p^2-1}\right)\prod _{\begin{array}{c}{p}^2|q\\ p\nmid (N-a)\end{array}}\left(1+\frac{1}{p^2-1}\right)$$

$$\times \prod _{\begin{array}{c}{p}^2|q\\ p\mid \mid (N-a)\end{array}}\left(1+\frac{1}{p^2-1}\right).$$

$$N\sum _{\begin{array}{c}a=1\\ (a,N)=1\end{array}}^{\infty }\frac{\mu (a)}{\phi ([a^2,q])}\cdot \text{mathbbm}{1}_{(a^2,q)\mid (N-a)}+O_{\epsilon }\left(N\sum _{a>w}\frac{1}{a^2}+N^{1/2+\epsilon }w\right).$$

$${\mathfrak{S}}_{0,1}(N)$$

$$=\prod _{\begin{array}{c}p=2\\ p\nmid N\end{array}}^{\infty }\left(\frac{p^2-1}{p^2}-\frac{1}{p^2(p-1)}\right)$$

$$=\prod _{\begin{array}{c}p=2\\ p\nmid N\end{array}}^{\infty }\left(1-\frac{1}{p(p-1)}\right).$$

$${\Vert \sum _{i\in I}{\xi }_i{\varphi }_i^{\ast }\Vert }^2\le \sum _{i\in I}{M}_i|{\xi }_i{|}^2$$

$$[f|g]=\sum _{n\le N}f(n)\overline{g(n)}$$

$$[f|g]$$

$$=\sum _{\begin{array}{c}N=n_1+n_2\\ n_1\equiv a[q]\end{array}}\Lambda (n_1){\mu }^2(n_2)$$

$$\left[f-\frac{1}{2}\sum_{q\in\mathcal{Q}}\xi_{q}(f)\Delta_{q}^{*}\Bigg{|}g-\frac{1}{2}\sum_{q\in\mathcal{Q}}\xi_{q}(g)\Delta_{q}^{*}\right]$$

$$\displaystyle\Bigg{[}f-\frac{1}{2}\sum_{q\in\mathcal{Q}}\xi_{q}(f)\Delta_{q}^{*}\Bigg{|}g-$$

$$=[f|g]-\frac{1}{2}\sum _{q\in \mathcal{Q}}{\xi }_q(f)[{\Delta }_q^{\ast }|g]-\frac{1}{2}\sum _{q\in \mathcal{Q}}{\xi }_q(g)[f|{\Delta }_q^{\ast }]$$

$$+\frac{1}{4}\sum _{q_1,q_2\in \mathcal{Q}}{\xi }_{q_1}(f){\xi }_{q_2}(g)[{\Delta }_{q_1}^{\ast }|{\Delta }_{q_2}^{\ast }].$$

$$[f|g]=\sum _{q\in \mathcal{Q}}\frac{[f|{\Delta }_q^{\ast }][{\Delta }_q^{\ast }|g]}{M_q}+O(R).$$

$$\sum_{a=1}^{k}\mathbf{e}_{k}(ar)=\begin{cases}k&\mbox{if $r\equiv 0[k]$,}\\ 0&\mbox{otherwise.}\end{cases}$$

$$c_r(n)=\sum _{d\mid (r,n)}d\mu \left(\frac{r}{d}\right)=\frac{\mu (r/(r,n))}{\phi (r/(r,n))}\phi (r),$$

$$c_{p}(n)=\begin{cases}-1&\mbox{if $p\nmid n$,}\\ \varphi(p)&\mbox{if $p\mid n$,}\end{cases}\mbox{\hskip 5.69054pt and \hskip 5.69054pt }c_{p^{2}}(n)=\begin{cases}0&\mbox{if $p\nmid n$,}\\ -p&\mbox{if $p\mid\!\mid n$,}\\ \varphi(p^{2})&\mbox{if $p^{2}\mid n$.}\end{cases}$$

$$\sum _{\begin{array}{c}k\mid a\\ d\mid k\end{array}}\mu (k/d)=\text{mathbbm}{1}_{d=a}.$$

$$\sum _{\begin{array}{c}n\le N\\ n\equiv a[q]\end{array}}\Lambda (n)=\frac{N}{\phi (q)}+O_{A,B}(N{\mathcal{L}}^{-B})$$

Full text

1. Introduction

The Goldbach conjecture states that every even integer greater than two can be expressed as the sum of two primes. Although this remains an open problem due to the parity phenomenon, there are various progress and relaxations which contributes toward this direction.

Mordern sieve method can be traced back to the earlier works of Brun. In 1920, Brun [1] showed that every sufficiently large even integer can be written as the sum of two numbers which have together at most nine prime divisors. Later, the celebrated result of Chen [3] established that every sufficiently large even integer can be written as the sum of a prime and a number with at most two prime factors.

Initiated by Linnik [15] in 1953, he showed that every sufficiently large even integer can be written as a sum of two primes and at most $K$ powers of two, where $K$ is an absolute constant although non-explicit. Many authors had made $K$ explicit where the best result $K=12$ is due to Liu & Lü [17], improving the remarkable result $K=13$ by Heath-Brown & Puchta [11].

Another relaxation is the ternary Goldbach conjecture which states that all odd integer greater than five is the sum of three primes. Vinogradov [23] developed a way to estimate sums over primes which combined with the circle method showed the ternary Goldbach conjecture is true for all large odd integer greater than $C>0$. Recently, Helfgott [12] completed the proof of ternary Goldbach conjecture by sufficiently reducing the size $C$ and verified that no counterexample exists below $C$.

We also have results when we replace one of the primes in the Goldbach conjecture by a square-free integer. Estermann [6] obtained an asymptotic formula for the number of representation of a sufficently large integer as the sum of a prime and a square-free number. Later, Page [20] improved on the error term of Estermann [6] and Mirsky [18] improved and extended these results to count the number of representions of an integer as the sum of a prime and a $k$-free number. Recently, Dudek [5] by tools of explicit number theory demonstrated that every integer greater than two can be a sum of a prime and a square-free integer.

In this paper we are motivated by a question posed in the PhD thesis of Dudek [4, Chapter 6, Problem 8]. Specifically Dudek asked for $(a,q)=1$, can all sufficiently large integer without local obstruction be a sum of a prime $p$ such that $p\equiv a\pmod{q}$ and a square-free integer. A similar problem on the number of representations of a sufficiently large integer to be the sum of two square-free integer has been studied by Evelyn & Linfoot [8] and later simplified by Estermann [7]. The best result is achieved by Brüdern & Perelli [2].

There are mainly two advanced techniques for attacking certain binary additive problems: sieve method [9] and the dispersion method of Linnik [16]. We note that the circle method [22] has certain difficulity in dealing with binary additive problems. Our method applied here is due to Ramaré [21] on his notion of local model, and can be viewed as an abstract circle method. We remark that Heath-Brown [10] had already notice this connection for his alternate prove of Vinogradov’s three prime theorem [23].

Lastly the techniques used here may be adapted for various other binary additive problems. In particular, the author expects that it should be possible to prove an asymptotic bound for the number of representation of an integer as the sum of a square-free integer and a prime $p$ such that $p+1$ is square-free.

2. Notation

The statements $U=O(V)$ and $U\ll V$ are both equivalent to $|U|\leq c|V|$ for some fixed positive constant $c$. If $c$ depends on a parameter, say $a$, then we may write $U=O_{a}(V)$ or $U\ll_{a}V$.

For completeness, we recall the following standard notation in analytic number theory.

[TABLE]

3. Main result

We denote

[TABLE]

to be the weighted number of representations for $N$ as the sum of a prime congruent to $a$ modulo $q$ and a square-free integer.

We now state a bound for $\mathcal{R}_{a,q}(N)$ which is uniform for small $q$.

Theorem 3.1**.**

Let $C_{1},C_{2}>0$ then we have

[TABLE]

uniformly for $(a,q)=1$ and $q\leq\mathcal{L}^{C_{1}}$. The singular series is given by

[TABLE]

The implied constant in the reminder term is ineffective as the Siegel-Walfisz Theorem is ineffective. In view of Lemma 5.3, we can rewrite the singular series in a more rudimentary form

[TABLE]

If $p_{1}^{2}\mid q$ and $p_{1}^{2}\mid(N-a)$ then it follows for all primes $p\equiv a[q]$ that $N-p$ is never square-free and hence $\mathcal{R}_{a,q}(N)=0$. This coincide exactly to the case when $\mathfrak{S}_{a,q}(N)$ vanishes.

Using the well-known convolution identity $\mu^{2}(n)=\sum_{d^{2}|n}\mu(d)$ and assuming the GRH (Generalised Riemann Hypothesis), the error term in  (3.1) can be replaced by $O_{\varepsilon}(N^{3/4+\varepsilon})$. Indeed on the GRH, we obtain

[TABLE]

Taking $w=N^{1/4}$ gives the result. It seems the error term cannot be further improved (assuming the GRH) using the local model method.

Observe that when we take $q=1$ in Theorem 3.1, we obtain a special case of Mirsky [18] (after weighing but with a weaker error term). Indeed our singular series simplifies to

[TABLE]

4. Outline of Method

For a more thorough exposition, see [21, Chapter 1,4 &17]. Our method will model that of [21, Chapter 19] where therein Ramaré proved an asymptotic bound for the number of representations of a sufficiently large integer as a sum of two square-free integer.

We press forward and recall the definition of local and global product, see (5.5) and (4.1) respectively.

Set $\mathcal{H}$ as an almost orthogonal system by the following collection of information:

1. (i)

a finite family $(\phi_{i}^{*})_{i\in I}$ of elements of $\mathcal{H}$, 2. (ii)

a finite family $(M_{i})_{i\in I}$ of positive real numbers,

and

[TABLE]

for all $(\xi_{i})_{i\in I}\in\mathbb{C}^{I}$.

The special case of choosing an orthogonal basis is enlightening. If $(\phi_{i}^{*})_{i\in I}$ were orthogonal then we may take $M_{i}=\lVert\phi_{i}^{*}\rVert^{2}$.

Let $\mathscr{F}(\mathbb{N})$ be the vector space of all complex valued functions on the positive integers and

[TABLE]

be the usual scalar product (global Hermitian product) for all $f,g\in\mathscr{F}(\mathbb{N})$.

Let $(a,q)=1$, $f(n)=\Lambda(n)\mathbbm{1}_{n\equiv a[q]}$ and $g(n)=\mu^{2}(N-n)$. Observe that

[TABLE]

is the weighted number of representations for $N$ to be a sum of a prime congruent to $a$ modulo $q$ and a square-free integer. Our ultimate goal is to compute $[f|g]$ and we shall use the notion of local model to this end.

Indeed let $\mathcal{Q}\subseteq\mathbb{N}$ be a carefully chosen set of moduli, we construct two local models $(\rho_{q}^{*})_{q\in\mathcal{Q}}$ and $(\gamma_{q}^{*})_{q\in\mathcal{Q}}$ to approximate $f$ and $g$ respectively, and in some sense they are made to copy the distribution of $f$ and $g$ in arithmetic progression respectively.

Next we take $(\frac{1}{2}\Delta_{q}^{*})_{q\in\mathcal{Q}}$ to be essentially the union of some linear combination of $(\rho_{q}^{*})_{q\in\mathcal{Q}}$ and $(\gamma_{q}^{*})_{q\in\mathcal{Q}}$, this will be the local model accountable for both $f$ and $g$. Furthermore for all $q_{1},q_{2}\in\mathcal{Q}$, we set $M_{q_{1}}=\sum_{t\in\mathcal{Q}}|[\Delta_{t}^{*}|\Delta_{q_{1}}^{*}]|$, see [21, Lemma 1.1]. This gives rise to an almost orthogonal system and in particular will imply the scalar product

[TABLE]

is small in a suitable sense. The construction will also secure $[\Delta_{q_{1}}^{*}|\Delta_{q_{2}}^{*}]$ to be small when $q_{1}\neq q_{2}$. Expanding the inner product, we have

[TABLE]

Here we take $\xi_{q}(f)=[f|\Delta_{q}^{*}]/M_{q}$ and $\xi_{q}(g)=[g|\Delta_{q}^{*}]/M_{q}$ as motivated by the orthogonal case. Simplifying gives

[TABLE]

The error term can be shown to be sufficiently small by appealing to the local model. The summand in the sum above can then be replaced by a tractable expression for which we can compute explicitly and the result soon follows.

5. Preparations

5.1. Number theoretical considerations

We record here various number theoretical lemmas needed in subsequent sections. For completeness we will also include several straightforward lemmas that may be applied freely without reference.

First we recall the well-known orthogonality of exponential sums [19, Equation 4.1].

Lemma 5.1**.**

For any positive integer $k$, we have

[TABLE]

Next we recall the Chinese remainder theorem for arbitrary modulus [14, Theorem 3.12].

Lemma 5.2** (Chinese remainder theorem).**

Let $a_{i},m_{i}\in\mathbb{N}$ for $i=1,\ldots,n$ and $L=[m_{1},\ldots,m_{n}]$. The following system of congruences $x\equiv a_{i}[m_{i}]$ for $i=1,\ldots,n$ is solvable if and only if $a_{i}\equiv a_{j}[(m_{i},m_{j})]$ for any $1\leq i,j\leq n$. If the system is solvable then $x\equiv\sigma[L]$ for some $1\leq\sigma\leq L$and any two such $x$ are congruent modulo $L$. Moreover, $(a_{i},m_{i})=1$ for $1\leq i\leq n$ if and only if $(\sigma,L)=1$.

We recall from [19, Theorem 4.1] some fundamental properties of Ramanujan sums.

Lemma 5.3**.**

For any positive integers $n$ and $r$, the Ramanujan sum $c_{r}(n)$ is a multiplicative function of $r$. Moreover we have

[TABLE]

and hence $|c_{r}(n)|=\varphi((r,n))$. In particular

[TABLE]

The next result provides an explicit expression for detecting equality for divisors [21, Corollary 3.1].

Lemma 5.4**.**

For integer $a\geq 1$ and any divisor $d$ of $a$, we have

[TABLE]

We record below a sensational result which gives an estimate for the number of primes in an arithmetic progression for small moduli [13, Corollary 5.29].

Lemma 5.5** (Siegel-Walfisz).**

For any $A,B>0$, we have

[TABLE]

uniformly for $(a,q)=1$ and $q\leq\mathcal{L}^{A}$.

Finally we state an auxiliary lemma that we will need later.

Lemma 5.6**.**

For all cubefree positive integers $m$ and $n$, we have

[TABLE]

Proof.

Write $n=n_{1}n_{2}^{2}$ and $m=m_{1}m_{2}^{2}$ where $\mu^{2}(n_{1}n_{2})=\mu^{2}(m_{1}m_{2})=1$. We factor our sum

[TABLE]

If there exist a prime factor that divides $n$ but not $m$ then the sum vanishes. By symmetry, we are left to consider the case $m=n$ and in turn the sum simplifies to

[TABLE]

\sqcap$$\sqcup

5.2. Arithmetic functions in arithmetic progression

The following result provides an asymptotic for the number of square-free integers in an arithmetic progression.

Note that we canonically extend the characteristics function of the square-free integers to negative integers by $\mu^{2}(-n)=\mu^{2}(|n|)$. We recall a result from [21, Lemma 19.1].

Lemma 5.7**.**

For any $\varepsilon>0$, we have

[TABLE]

Here

[TABLE]

As in the notation of [21, Chapter 19], for cubefree $q$ we define

[TABLE]

with $a\in\mathbb{Z}/q\mathbb{Z}$ and

[TABLE]

Note that in some sense $\gamma_{q}^{*}(N-n)$ is defined to imitate $\mu^{2}(N-n)$ in arithmetic progression.

We recall the following result from [21, Lemma 19.2] which provides an explicit expression for $\gamma_{q}^{*}$.

Lemma 5.8**.**

We have $\gamma_{q}^{*}(a)=6t(q)c_{q}(a)/\pi^{2}$ if $q$ is a positive cubefree integer, while if $q$ has a cubic factor greater than $1$ then $\gamma_{q}^{*}(a)=0$.

Let $(a^{\prime},q^{\prime})=1$ and $q$ be a positive cubefree integer, denote

[TABLE]

To motivate the definition of $\rho_{q}$, consider the sum

[TABLE]

By the Chinese remainder theorem, the simultaneous congruence equations is solvable if and only if $(a,q)=1$ and $(q,q^{\prime})\mid(a-a^{\prime})$. If this is the case then for some $0\leq\sigma<[q,q^{\prime}]$ we should expect

[TABLE]

We remark that if $a_{1}\equiv a_{2}[q]$ then $\rho_{q}(a_{1})=\rho_{q}(a_{2})$.

Lemma 5.9**.**

For $q_{1},q_{2}$ positive cubefree integers with $(q_{1},q_{2})=1$, we have

[TABLE]

In particular for any positive integer $a$, $\varphi(q^{\prime})\rho_{q}(a)$ is a multiplicative function of $q$.

Proof.

If $(a,q_{1}q_{2})>1$ then either $(a,q_{1})>1$ or $(a,q_{2})>1$. If $(q_{1}q_{2},q^{\prime})\nmid(a-a^{\prime})$ then we have $(q_{1},q^{\prime})\nmid(a-a^{\prime})$ or $(q_{2},q^{\prime})\nmid(a-a^{\prime})$. Consequently for both cases we obtain

[TABLE]

Clearly $(q_{1}q_{2},q^{\prime})\mid(a-a^{\prime})$ and $(a,q_{1}q_{2})=1$ if and only if we satisfy both the conditions

[TABLE]

Hence it is enough to show

[TABLE]

Let us recall the identity $ab=[a,b]\cdot(a,b)$ for all positive integers $a,b$. Consequently

[TABLE]

since

[TABLE]

The result follows immediately.     \sqcap$$\sqcup

We are now ready to define our local approximation for $f$. Set

[TABLE]

for any positive cubefree integer $q$.

The next result provides an explicit expression for $\rho_{q}^{*}$.

Lemma 5.10**.**

Let $q=q_{1}q_{2}^{2}$ with $\mu^{2}(q_{1}q_{2})=1$. If $a_{1}\equiv a_{2}[q]$ then $\rho_{q}^{*}(a_{1})=\rho_{q}^{*}(a_{2})$. For any positive integer $a$, $\varphi(q^{\prime})\rho_{q}^{*}(a)$ is a multiplicative function of $q$. Furthermore

[TABLE]

Proof.

We note that if $a_{1}\equiv a_{2}[q]$ then $a_{1}\equiv q_{2}[d]$ for all divisors $d$ of $q$. Hence

[TABLE]

Appealing to Lemma 5.9, $\varphi(q^{\prime})\rho_{q}^{*}$ is a Dirichlet convolution of two multiplicative functions, hence $\varphi(q^{\prime})\rho_{q}^{*}$ is multiplicative in $q$. Therefore we factor

[TABLE]

We first consider $\varphi(q^{\prime})\rho_{q_{1}}^{*}$. Recall that $\rho_{1}(a)=1/\varphi(q^{\prime})$ and so

[TABLE]

From (5.2), we check

[TABLE]

In the last line, we note that the condition implies that $(a,p)=1$ since $(a^{\prime},q^{\prime})=1$. Therefore

[TABLE]

Lastly, we turn to $\varphi(q^{\prime})\rho_{q_{2}^{2}}^{*}$ and we see for any $p^{2}\mid q_{2}^{2}$ that

[TABLE]

We readily check that $\varphi(q^{\prime})\rho_{p^{2}}^{*}(a)$ is

[TABLE]

We condense the expression to the simpler

[TABLE]

since $(a^{\prime},q^{\prime})=1$. Hence we obtain

[TABLE]

The result follows immediately.     \sqcap$$\sqcup

5.3. Local Hermitian product

In this section we compute various local Hermitian products explicitly.

We denote

[TABLE]

for all positive cubefree integer $q$.

For fixed $q$, we denote $\mathscr{F}(\mathbb{Z}/q\mathbb{Z})$ to be the vector space of complex valued functions over $\mathbb{Z}/q\mathbb{Z}$. We endow this vector space with the local Hermitian product by setting

[TABLE]

for all $f,g\in\mathscr{F}(\mathbb{Z}/q\mathbb{Z})$.

We now state an explicit expression for the norms of $\vartheta_{q}^{*}$ and $\rho_{q}^{*}$.

Lemma 5.11**.**

For all $q=q_{1}q_{2}^{2}$ with $\mu^{2}(q_{1}q_{2})=1$, we have

[TABLE]

and

[TABLE]

Proof.

The expression for $\lVert\vartheta_{q}^{*}\rVert_{q}^{2}$ can be derived as in [21, Equation (19.10)].

Write

[TABLE]

If $q_{2}^{2}\nmid q^{\prime}$ then we are done since $\rho_{q}^{*}=0$ by Lemma 5.10. Otherwise, by the Chinese remainder theorem we factor

[TABLE]

Appealing to (5.3), we readily check

[TABLE]

By Lemma 5.10, expanding the Ramanujan sum and interchanging the summation, we obtain

[TABLE]

By orthogonality, we get

[TABLE]

It follows

[TABLE]

since

[TABLE]

\sqcap$$\sqcup

For all positive cubefree integer $q$, we denote

[TABLE]

where

[TABLE]

We now state a result which provides an explicit expression for $b(p)$ and $b(p^{2})$.

Lemma 5.12**.**

The function $b(q)$ is a multiplicative function of $q$. Moreover we have

[TABLE]

and

[TABLE]

Proof.

For positive cubefree integers $q_{1},q_{2}$ with $\mu^{2}(q_{1}q_{2})=1$, we have by the Chinese remainder theorem

[TABLE]

where in the second line we used Lemma 5.10. Similarly, we show that $b(q)$ is a multiplicative function in $q$.

First we assume that $(p,q^{\prime})=1$, appealing to (5.3) we get

[TABLE]

Since $(r,p)=1$ and hence by orthogonality we have

[TABLE]

Now assume $p\mid q^{\prime}$ and appealing to (5.3) we obtain

[TABLE]

Since $(r,p)=1$, orthogonality gives

[TABLE]

Next we consider $b(p^{2})$. If $p^{2}\nmid q^{\prime}$ then we are done, otherwise expanding the Ramanujan sum and interchanging summation, we obtain

[TABLE]

Appealing to orthogonality then gives

[TABLE]

\sqcap$$\sqcup

Next, we derive an explicit expression for the cross product $[\vartheta_{q}^{*}|\rho_{q}^{*}]_{q}$.

Lemma 5.13**.**

For all $q=q_{1}q_{2}^{2}$ with $\mu^{2}(q_{1}q_{2})=1$, we have

[TABLE]

Proof.

If $q_{2}^{2}\nmid q^{\prime}$ then we are done, otherwise

[TABLE]

Therefore by the previous lemma, we arrive at

[TABLE]

and the result follows.     \sqcap$$\sqcup

5.4. Local models and their products

Let $q=q_{1}q_{2}^{2}$ with $\mu^{2}(q_{1}q_{2})=1$, we denote

[TABLE]

[TABLE]

where

[TABLE]

is just $\rho_{q}^{*}$ but without the divisibility condition on $q^{\prime}$, see Lemma 5.10. In particular, we can write

[TABLE]

Below, we compute the norms of $\eta_{q}^{*}$ and $\kappa_{q}^{*}$.

Lemma 5.14**.**

We have

[TABLE]

[TABLE]

and

[TABLE]

Proof.

The norms for $\eta_{q}^{*}$ and $\kappa_{q}^{*}$ follows immediately from Lemma 5.11, 5.13 and that both $\vartheta_{q}^{*}$ and $\rho_{q}^{*}$ are real valued.

Showing

[TABLE]

also follows from Lemma 5.11, 5.13.     \sqcap$$\sqcup

Note that $\lVert\eta_{q}^{*}\rVert_{q}^{2}\neq 0$ and $\lVert\kappa_{q}^{*}\rVert_{q}^{2}=0$ when

[TABLE]

Next, we will show there is a Hermitian relationship between the global (4.1) and local (5.5) products.

Given a function $h\in\mathscr{F}(\mathbb{Z}/q\mathbb{Z})$, we denote $\nabla_{q}(h)$ to be a function on $[1,N]$ defined by

[TABLE]

For a function $j\in\mathscr{F}([1,N])$, we denote $\Delta_{q}(j)$ to be a function on the positive integers such that

[TABLE]

Note that $\Delta_{q}\nabla_{q}$ is not the identity. For $h_{1}\in\mathscr{F}([1,N])$ and $h_{2}\in\mathscr{F}(\mathbb{Z}/q\mathbb{Z})$, we readily check that

[TABLE]

Indeed

[TABLE]

Let us set $\phi_{q}^{*}=\nabla_{q}\eta_{q}^{*}$ and $\psi_{q}^{*}=\nabla_{q}\kappa_{q}^{*}$. Now we compute the cross products of $\phi_{q}^{*}$ and $\psi_{q}^{*}$.

Lemma 5.15**.**

For all positive cubefree integers $m$ and $n$, we have

[TABLE]

Proof.

Write $m=m_{1}m_{2}^{2}$ and $n=n_{1}n_{2}^{2}$ with $\mu^{2}(m_{1}m_{2})=\mu^{2}(n_{1}n_{2})=1$. By appealing to (5.4), (5.6), Lemma 5.8 and 5.10, we expand

[TABLE]

where

[TABLE]

Using the explicit form of $\vartheta_{q}^{*}$ and applying Lemma 5.3, we have

[TABLE]

since $[d_{1},d_{2}](d_{1},d_{2})=d_{1}d_{2}$. By Lemma 5.6, we get

[TABLE]

Next we deal with $S_{4}$. Expanding each for the four Ramanujan sum and taking the summation over $n$ inside, we obtain

[TABLE]

Observe that $(d_{1},d_{3})=(d_{1},d_{4})=(d_{2},d_{3})=(d_{2},d_{4})=1$ since $m_{2}^{2}\mid q^{\prime}$ and $n_{2}^{2}\mid q^{\prime}$. By the Chinese reminder theorem, the system of congruences in the summation over $n$ can be reduced to one congruence $n\equiv\sigma[[d_{1},d_{2},d_{3},d_{4}]]$ for some $\sigma$. Since $(d_{1},d_{3})=(d_{2},d_{4})=1$, we verify that

[TABLE]

Next, we glue the variables $d_{1},d_{3}$ and $d_{2},d_{4}$ together and get

[TABLE]

By Lemma 5.6, we obtain

[TABLE]

Now we consider $S_{2}$. Again, expanding the Ramanujan sums and interchanging summations, we get

[TABLE]

Since $(d_{2},d_{3})=1$, the system of congruence in the inner sum is solvable if and only if $(d_{1},d_{3})\mid(N-a^{\prime})$ and $(d_{1},d_{2})\mid N$. It follows by the Chinese remainder theorem that

[TABLE]

where

[TABLE]

The dependency on the divisibility condition over the summation of $d_{2}$ and $d_{3}$ is troublesome. We deal with this by Lemma 5.4 and observe that

[TABLE]

and

[TABLE]

Substituting this into $M$ and gluing the variables $d_{2}$ and $d_{3}$ and noticing that $(k,k^{\prime})=1$ since $(d_{2},d_{3})=1$, $k^{\prime}\mid d_{2}$ and $k\mid d_{3}$, we assert

[TABLE]

If $m\neq n$ then $M=0$ by Lemma 5.6. If $m=n$ then again by Lemma 5.6, we get

[TABLE]

Notice that for $k^{\prime}\mid\frac{n_{1}}{(n_{1},q^{\prime})}$ and $k\mid(n_{1},q^{\prime})n_{2}^{2}$, we rewrite the Euler totient function as a Dirichlet convolution

[TABLE]

Substituting this into the above and interchanging summation, we obtain

[TABLE]

From Lemma 5.3 we notice

[TABLE]

and

[TABLE]

Therefore the sum $M$ in question collapses into

[TABLE]

Hence we have

[TABLE]

Similarly we also have

[TABLE]

Gathering all the estimates above, we finally get

[TABLE]

and the result follows from Lemma 5.14.

The remaining bounds for $[\psi_{m}^{*}|\psi_{n}^{*}],[\psi_{m}^{*}|\phi_{n}^{*}]$ and $[\phi_{m}^{*}|\psi_{n}^{*}]$ follows immediately from our computation of $S_{1},S_{2},S_{3},S_{4}$.     \sqcap$$\sqcup

5.5. Approximating $f$ and $g$

In this section we impose implicitly the condition

[TABLE]

Take our set of moduli to be

[TABLE]

Here

[TABLE]

where $D_{1},D_{2}\geq C_{1}$ will be chosen later. We also set

[TABLE]

and in particular this implies $q^{\prime}\leq Q\ll_{\varepsilon}N^{\varepsilon}$. The set $\mathcal{Q}$ will be referenced in the preceding lemmas.

Finally, we recall $(a^{\prime},q^{\prime})=1$ and let

[TABLE]

and

[TABLE]

for all $n\leq N$.

Lemma 5.16**.**

For any $A>0$ and $q\in\mathcal{Q}$, we have

[TABLE]

and

[TABLE]

In particular, we have

[TABLE]

and

[TABLE]

All implied constant above may depend on $A,C_{1},D_{1},D_{2},\varepsilon$.

Proof.

By (5.8), we get

[TABLE]

Note that by (5.6) and (5.7), we have

[TABLE]

By the Möbius inversion formula that

[TABLE]

and it follows

[TABLE]

Following the proof of Lemma 5.15 the summand is zero unless $d=q$, and hence $[\eta_{q}^{*}|\rho_{q}]_{q}=[\eta_{q}^{*}|\rho_{d}^{*}]_{q}$. It follows

[TABLE]

Write

[TABLE]

where

[TABLE]

We crudely bound using the Siegel-Walfisz Theorem (Lemma 5.5) to get

[TABLE]

Next we turn to $[\phi_{q}^{*}|g]$. Similarly we get

[TABLE]

Note that by (5.6) and (5.7), we have

[TABLE]

By the Möbius inversion formula, we have

[TABLE]

and it follows

[TABLE]

Again following the proof of Lemma 5.15, the summand vanishes unless $d=q$, and hence $[\eta_{q}^{*}|\vartheta_{q}]_{q}=[\eta_{q}^{*}|\vartheta_{d}^{*}]_{q}$. Therefore

[TABLE]

We can deal with $[\eta_{q}^{*}|\Delta_{q}g-N\vartheta_{q}]_{q}$ just like above but we apply Lemma 5.7 instead of Lemma 5.5 to get

[TABLE]

The bounds for $[\psi_{q}^{*}|f]$ and $[\psi_{q}^{*}|g]$ follows similarly.     \sqcap$$\sqcup

Denote

[TABLE]

and note that $\lVert\kappa_{q}^{*}\rVert_{q}=0$ if and only if $q\in\mathcal{E}$. The set $\mathcal{E}$ will be referenced in the preceding lemmas.

Now we give upper and lower bounds for these norms.

Lemma 5.17**.**

For all $q=q_{1}q_{2}^{2}\in\mathcal{Q}$, we have

[TABLE]

The same holds when we replace $(\eta_{q}^{*},\mathcal{Q})$ with $(\kappa_{q}^{*},\mathcal{Q}\backslash\mathcal{E})$.

Proof.

From Lemma 5.14, we get

[TABLE]

Clearly

[TABLE]

and the upper bound follows.

For the lower bound, note that

[TABLE]

strictly divides $\varphi(q)$, hence

[TABLE]

The bound for $\lVert\kappa_{q}^{*}\rVert_{q}^{2}$ is similar.     \sqcap$$\sqcup

Next we need to estimate the sums

[TABLE]

for all $q\in\mathcal{Q}$ and

[TABLE]

for all $q\in\mathcal{Q}$. But first, we need an a priori bound.

Lemma 5.18**.**

We have

[TABLE]

Proof.

The sum is majorized by

[TABLE]

\sqcap$$\sqcup

We only show for (5.14) as (5.15) is similar. By Lemma 5.15 and 5.18, we get that (5.14) is

[TABLE]

This motivates the following definition. Let $\varepsilon>0$ and $C=C(\varepsilon)>0$ be large enough so that it dominates both (5.14) and (5.15). Next we set

[TABLE]

for all $q\in\mathcal{Q}$ and

[TABLE]

for all $q\in\mathcal{Q}\backslash\mathcal{E}$.

The following result shows that we can replace $M(\phi_{q}^{*})$ and $M(\psi_{q}^{*})$ by $N\lVert\eta_{q}^{*}\rVert_{q}^{2}$ and $N\lVert\kappa_{q}^{*}\rVert_{q}^{2}$ respectively in the summand up to an error term.

Lemma 5.19**.**

If

[TABLE]

for all $q\in\mathcal{Q}$ or if

[TABLE]

for all $q\in\mathcal{Q}$ then

[TABLE]

The same holds if we replace $(\phi_{q}^{*},\eta_{q}^{*},\mathcal{Q})$ by $(\psi_{q}^{*},\kappa_{q}^{*},\mathcal{Q}\backslash\mathcal{E})$.

Proof.

Taking the difference and by Lemma 5.17, it is enough to bound

[TABLE]

First we suppose (5.16) and recalling (5.1), we majorize the sum above by

[TABLE]

Therefore (5.18) holds. The same argument holds if we replace $(\phi_{q}^{*},\eta_{q}^{*},\mathcal{Q})$ by $(\psi_{q}^{*},\kappa_{q}^{*},\mathcal{Q}\backslash\mathcal{E})$.

Next we assume (5.17). Using (5.19) and Lemma 5.17, we are lead to bound

[TABLE]

Recalling (5.1), the sum above is majorized by

[TABLE]

The same holds when we replace $(\phi_{q}^{*},\eta_{q}^{*},\mathcal{Q})$ by $(\psi_{q}^{*},\kappa_{q}^{*},\mathcal{Q}\backslash\mathcal{E})$.     \sqcap$$\sqcup

Lemma 5.20**.**

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

[TABLE]

is

[TABLE]

Proof.

Write

[TABLE]

where $\beta_{q}=M(\phi_{q}^{*})^{-1}[g|\phi_{q}^{*}]$. By Lemma 5.16, we replace $[\phi_{q}^{*}|g]$ by $N[\eta_{q}^{*}|\vartheta_{q}^{*}]$ up to an error term. Indeed

[TABLE]

after recalling (5.1) and using the bound

[TABLE]

collected from Lemma 5.16 and 5.17. Reiterating again we have

[TABLE]

We repeat this for the other sum and in total we get

[TABLE]

By Lemma 5.19, we replace $M(\phi_{q}^{*})$ by $N\lVert\eta_{q}^{*}\rVert_{q}^{2}$ up to an error term and get

[TABLE]

We recall from (5.13)

[TABLE]

it follows

[TABLE]

by (5.12), (5.13) and Lemma 5.11, 5.14.

Next, we reiterate the process and replace $M(\psi_{q}^{*})$ by $N\lVert\kappa_{q}^{*}\rVert_{q}^{2}$ up to an error term. In total we get

[TABLE]

By Lemma 5.11 we have

[TABLE]

and thus completing the series we obtain

[TABLE]

Observe that by Lemma 5.7

[TABLE]

and the result follows.     \sqcap$$\sqcup

Lemma 5.21**.**

The sum

[TABLE]

Proof.

By Lemma 1.1 and 1.2 of [21] we see that

[TABLE]

Therefore it is enough to bound $[f|f]$ and indeed we have

[TABLE]

\sqcap$$\sqcup

Lemma 5.22**.**

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

[TABLE]

is

[TABLE]

where $\mathfrak{S}_{a^{\prime},q^{\prime}}(N)$ is the singular series defined in Theorem 3.1.

Proof.

For simplicity all implied constant in the $O(\cdot)$ term may depend on $A,C_{1},D_{1},D_{2},\varepsilon$.

Set $\alpha_{q}=M(\phi_{q}^{*})^{-1}[f|\phi_{q}^{*}]$. By Lemma 5.16, we obtain

[TABLE]

by using the bound

[TABLE]

provided by Lemma 5.16 and 5.17.

Now set $\beta_{q}=M(\phi_{q}^{*})^{-1}N[\eta_{q}^{*}|\vartheta_{q}^{*}]_{q}$. Then again by Lemma 5.16, we have

[TABLE]

By Lemma 5.16 and 5.17, we assert

[TABLE]

and recalling (5.1) we get that the error term is $O\left(NQ^{8}\mathcal{L}^{-A}\right)$.

We do the same for the other sum and we ultimately get

[TABLE]

Next by applying Lemma 5.19 we replace $M(\phi_{q}^{*})$ with $N\lVert\eta_{q}^{*}\rVert_{q}^{2}$ up to an error term and obtain

[TABLE]

We recall from (5.13) and (5.12) that

[TABLE]

and

[TABLE]

It follows

[TABLE]

by (5.12), (5.13) and Lemma 5.13, 5.14.

We reiterate the same procedure and replace $M(\psi_{q}^{*})$ with $N\lVert\kappa_{q}^{*}\rVert_{q}^{2}$ up to an error term. In total we have

[TABLE]

Next we compute the sum

[TABLE]

We complete the series and so the right hand side becomes

[TABLE]

up to an error term of

[TABLE]

since the number of divisors of $q^{\prime}$ is at most $O_{\varepsilon}(Q^{\varepsilon})$ and by recalling (5.1). The singular series can be represented as an infinite product

[TABLE]

If $p\nmid q^{\prime}$ then the term in the product simplifies to

[TABLE]

If $p\mid\!\mid q^{\prime}$ then the term in the product simplifies to

[TABLE]

If $p^{2}\mid q^{\prime}$ then the term in the product simplifies to

[TABLE]

The result follows.     \sqcap$$\sqcup

6. Proof of Theorem 3.1

For any $h_{1},h_{2}\in\mathscr{F}([1,N])$, let us denote

[TABLE]

We recall $f,g$ from (5.10), (5.11) respectively and consequently the scalar product

[TABLE]

By Cauchy’s inequality, we assert

[TABLE]

By Lemma 5.20 and 5.21, the right hand side is majorised by

[TABLE]

Hence we can approximate $[f|g]$ by $\langle f|g\rangle$ and by Lemma 5.22 we obtain

[TABLE]

Recall from (5.9) that $Q_{1}=\mathcal{L}^{D_{1}},Q_{2}=\mathcal{L}^{D_{2}}$, and $Q=\max\{Q_{1},Q_{2}\}$. Taking

[TABLE]

simplifies the error term to

[TABLE]

Notice that

[TABLE]

The double sum can be estimated crudely by

[TABLE]

and consequently

[TABLE]

If $\mathfrak{S}_{a^{\prime},q^{\prime}}(N)=0$ then there exists $p^{2}\mid q^{\prime}$ such that $p^{2}\mid(N-a)$ by our remark after Theorem 3.1. Hence $\mathcal{R}_{a^{\prime},q^{\prime}}(N)=0$ and the result follows immediately. Otherwise if $\mathfrak{S}_{a^{\prime},q^{\prime}}(N)\neq 0$ then we bound from below

[TABLE]

Applying a Taylor series expansion for the logarithm

[TABLE]

valid for $|x|<1$, we obtain

[TABLE]

Therefore

[TABLE]

and so

[TABLE]

Acknowledgement

The author thanks I. E. Shparlinski and L. Zhao for many helpful comments and discussions, and O. Ramaré for helpful discussions. The author also thanks the referee for many helpful comments and suggestions. This work is supported by an Australian Government Research Training Program (RTP) Scholarship, UNSW Science PhD Writing Scholarship, and the Lift-off Fellowship of the Australian Mathematical Society.