¹¹footnotetext: Zhen Guo is the corresponding author.
Keywords: Divisor function; exponential sum; double large sieve.
MR(2020) Subject Classification: 11L07, 11L20, 11N37.

On a sum of the error term of the Dirichlet divisor function over primes

Zhen Guo & Xin Li Department of Mathematics, China University of Mining and Technology, Beijing, 100083, People’s Republic of China [email protected] Department of Mathematics, China University of Mining and Technology, Beijing, 100083, People’s Republic of China [email protected]

Abstract.

Let $d(n)$ be the Dirichlet divisor function and $\Delta(x)$ denote the error term of the sum $\sum_{n\leqslant x}d(n)$ for a large real variable $x$ . In this paper we focus on the sum $\sum_{p\leqslant x}\Delta^{2}(p)$ , where $p$ runs over primes. We prove that there exists an asymptotic formula.

1. Introduction and Main Result

Let $d(n)$ be the Dirichlet divisor function and $\Delta(x)$ denote the error term of the sum $\sum_{n\leqslant x}d(n)$ for a large real variable $x$ . It was first proved by Dirichlet that $\Delta(x)=O(x^{1/2})$ . Over the years the exponent $1/2$ was improved by many authors[7, 13, 19, 20]. Until now the best result

\Delta(x)\ll x^{131/416}(\log x)^{26497/8320}

(1.1)

was given by Huxley[8].

It is conjectured that $\Delta(x)\ll x^{1/4+\varepsilon}$ , which is supported by the classical mean square result

\int_{1}^{T}\Delta^{2}(x)dx=\frac{\mathcal{C}}{6\pi^{2}}T^{3/2}+O(T^{5/4+\varepsilon})

(1.2)

for a large real number $T$ , and where

\mathcal{C}=\sum_{n=1}^{\infty}\frac{d^{2}(n)}{n^{3/2}}

is a constant. It was proved by Cramér[1] in 1922. The result was improved by Tong[17] in 1956, Preissmann[15] in 1988, and Lau and Tsang[14] in 2009.

Also, the discrete mean values

\mathcal{D}_{2}(x):=\sum_{n\leqslant x}\Delta^{2}(n)

have been investigated by various authors. Define the continuous mean values

\mathcal{C}_{2}(x):=\int_{1}^{x}\Delta^{2}(t)dt.

Many authors found that the discrete and the continuous mean value formulas are connected deeply with each other and studied the difference between $\mathcal{D}_{2}(x)$ and $\mathcal{C}_{2}(x)$ . Hardy[4] studied the difference between $\mathcal{D}_{2}(x)$ and $\mathcal{C}_{2}(x)$ and derived that $\mathcal{D}_{2}(x)=O(x^{3/2+\varepsilon})$ . These differences were also studied by Furuya [3], he gave the asymptotic formula

\mathcal{D}_{2}(x)=\mathcal{C}_{2}(x)+\frac{1}{6}x\log^{2}x+\frac{8\gamma-1}{12}x\log x+\frac{8\gamma^{2}-2\gamma+1}{12}x+\left\{\begin{array}[]{lr}O\\ \Omega_{\pm}\end{array}\right\}(x^{3/4}\log x).

In this paper, as an analogue of the discrete mean values, we consider a sum of $\Delta^{2}(\cdot)$ over a subset of $\mathbb{N}$ . More precisely over primes, namely $\sum_{p\leqslant x}\Delta^{2}(p)$ , where $p$ runs over primes and $x$ is a large real variable. And the result is stated as follows:

Theorem.

Let $p$ be prime and $x\geq 2$ is a large real number, then

\sum_{p\leqslant x}\Delta^{2}(p)=\frac{\mathcal{C}}{4\pi^{2}}\sum_{p\leqslant x}p^{1/2}+O(x^{23/16+\varepsilon})

(1.3)

holds for any $\varepsilon>0$ , where $\mathcal{C}$ is defined in (1.2).

2. Priliminary

2.1. Notations

Throught this paper, $\varepsilon$ denotes a sufficiently small positive number, not necessarily the same at each occurrence. Let $p$ always denote a prime number. As usual, $d(n)$ $\Lambda(n)$ and $\mu(n)$ denote the Dirichlet divisor function, the von Mangoldt function and the Möbius function, respectively, $\mathbb{C}$ denotes the set of complex numbers, $\mathbb{R}$ denotes the set of real numbers, and $\mathbb{N}$ denotes the set of natural numbers. We write $\mathbf{e}(t):=exp(2\pi it)$ . The notation $m\sim M$ means $M<m\leqslant 2M$ , $f(x)\ll g(x)$ means $f(x)=O(g(x))$ ; $f(x)\asymp g(x)$ means $f(x)\ll g(x)\ll f(x)$ . $\gcd(m_{1},\cdots,m_{j})$ denotes the greatest common divisor of $m_{1},\cdots,m_{j}$ $\in\mathbb{N}$ ( $j=2,3,\cdots$ ).

2.2. Auxiliary Lemmas

We need the following lemmas.

Lemma 2.1.

For real numbers $N$ and $x$ satisfying $1\ll N\ll x$ , we have

\Delta(x)=\delta_{1}(x,N)+\delta_{2}(x,N),

where

\delta_{1}(x,N)=\frac{x^{1/4}}{\sqrt{2}\pi}\sum_{n\leqslant N}\frac{d(n)}{n^{3/4}}\cos\left(4\pi\sqrt{nx}-\frac{\pi}{4}\right),

and $\delta_{2}(x,N)$ satisfies:

(i) $\delta_{2}(x,N)\ll x^{1/2}N^{-1/2}$ .

(ii) For any fixed real number $T\geqslant 2$ , we have

\int_{T}^{2T}\delta_{2}(x,N)dx\ll\frac{T^{3/2}\log^{3}T}{N^{1/2}}+T\log^{4}T.

Proof.

See, for example, the following references: Lau and Tsang[14], Tsang[18], Zhai[21]. ∎

Lemma 2.2.

Let $z\geqslant 1$ be a real number and $k\geq 1$ be an integer. Then for any $n\leqslant 2z^{k}$ , there holds

\Lambda(n)=\sum_{j=1}^{k}(-1)^{j-1}\binom{k}{j}\mathop{\sum\cdots\sum}_{\begin{subarray}{c}n_{1}n_{2}\cdots n_{2j}=n\\ n_{j+1},\cdots,n_{2j}\leqslant z\end{subarray}}(\log n_{1})\mu(n_{j+1})\cdots\mu(n_{2j}).

(2.1)

Proof.

See the argument on pp.1366-1367 of Heath-Brown[5]. ∎

Lemma 2.3.

Let $\mathscr{X}:=\{x_{r}\}$ , $\mathscr{Y}:=\{y_{s}\}$ be two finite sequence of real numbers with

|x_{r}|\leqslant X,\qquad|y_{s}|\leqslant Y,

and $\varphi_{r},\psi_{s}\in\mathbb{C}$ . Defining the bilinear forms

{\mathscr{B}}_{\varphi\psi}(\mathscr{X},\mathscr{Y})=\sum_{r\sim R}\sum_{s\sim S}\varphi_{r}\psi_{s}\mathbf{e}(x_{r}y_{s}),

then we have

|\mathscr{B}_{\varphi\psi}(\mathscr{X},\mathscr{Y})|^{2}\leqslant 20(1+XY)\mathscr{B}_{\varphi}(\mathscr{X},Y)\mathscr{B}_{\psi}(\mathscr{Y},X),

with

\mathscr{B}_{\varphi}(\mathscr{X},Y)=\sum_{|x_{r_{1}}-x_{r_{2}}|\leqslant Y^{-1}}|\varphi_{r_{1}}\varphi_{r_{2}}|

and $\mathscr{B}_{\psi}(\mathscr{Y},X)$ defined similarly.

Proof.

See Proposition 1 of Fouvry and Iwaniec[2]. ∎

Lemma 2.4.

Let $\{a_{i}\}$ and $\{b_{i}\}$ be two finite sequences of real numbers and let $\delta>0$ be a real number. Then we have

\#\{(r,s):|a_{r}-b_{s}|\leqslant\delta\}\leqslant 3(\#\{(r,r^{\prime}):|a_{r}-a_{r^{\prime}}|\leqslant\delta\})^{1/2}(\#\{(s,s^{\prime}):|b_{s}-b_{s^{\prime}}|\leqslant\delta\})^{1/2}

Proof.

See the argument of P266-267 in Zhai[22]. ∎

Lemma 2.5.

Let $\alpha\neq 0,1$ be a fixed real integer. For any integer $M\geqslant 2$ and any real number $\delta>0$ , we define $\mathcal{N}(M,\delta)$ as being the number of quadruplets

\underline{m}=(m_{1},m_{2},m_{3},m_{4})\in\{M+1,M+2,\cdots,2M\}^{4}

which satisfy the inequality

|{m_{1}}^{\alpha}+{m_{2}}^{\alpha}-{m_{3}}^{\alpha}-{m_{4}}^{\alpha}|\leqslant\delta M^{\alpha},

then we have

\mathcal{N}(M,\delta)\ll M^{2+\varepsilon}+\delta M^{4+\varepsilon},

where the implied constant depends only on $\varepsilon$ .

Proof.

See the argument on Theorem 2 of Robert and Sargos[16]. ∎

Lemma 2.6.

Let $N$ be a given large integer, $X$ be a real number and $\mathbf{N}$ denote the set $\{N+1,\cdots,2N\}$ . Suppose

v(\mathbf{N},X)=|\{n_{1},n_{2}\in\mathbf{N};|\sqrt{n_{1}}-\sqrt{n_{2}}|\leqslant(2X)^{-1}\}|,

then we have

v(\mathbf{N},X)\leqslant(1+2\sqrt{2N}X^{-1})|\mathbf{N}|.

Proof.

See Lemma 2 of Iwaniec and Sárközy[11]. ∎

Lemma 2.7.

Let $a,b$ are fixed real numbers and $a<b$ . Suppose $f(x)$ is a real function with $|f^{\prime}(x)|\leqslant 1-\theta$ and $f^{\prime\prime}(x)\neq 0$ on $[a,b]$ . We then have

\sum_{a<n\leqslant b}\mathbf{e}(f(n))=\int_{a}^{b}\mathbf{e}(f(x))dx+O(\theta^{-1}).

Proof.

See Lemma 8.8 of Iwaniec and Kowalski[10]. ∎

Lemma 2.8.

Let $F(x)$ be a real differentiable function such that $F^{\prime}(x)$ is monotonic and $|F^{\prime}(x)|\geqslant m>0$ for $a\leqslant x\leqslant b$ . Then

\left|\int_{a}^{b}e^{iF(x)}dx\right|\leqslant 4m^{-1}.

Proof.

See Lemma 2.1 of Ivić[9]. ∎

Lemma 2.9.

Suppose that $f(x):[a,b]\rightarrow\mathbb{R}$ has continuous derivatives of arbitrary order on $[a,b]$ , where $1\leqslant a<b\leqslant 2a$ . Suppose further that we have

|f^{\prime\prime}(x)|\asymp\lambda_{2},\qquad x\in[a,b].

Then

\sum_{a<n\leqslant b}\mathbf{e}(f(n))\ll a{\lambda_{2}}^{1/2}+{\lambda_{2}}^{-1/2}.

(2.2)

Proof.

See Corollary 8.13 of Iwaniec and Kowalski[10], or Theorem 5 of Chapter 1 in Karatsuba[12]. ∎

Lemma 2.10.

Let $T$ be a large real number and $H$ be a real number such that $1\leqslant H\leqslant T$ . We have

\displaystyle\int_{T}^{2T}\max_{h\leqslant H}(\Delta(t+h)-\Delta(t))^{2}dt\ll HT\log^{5}T.

Proof.

See Heath-Brown and Tsang[6]. ∎

Lemma 2.11.

Suppose $N_{1},N_{2}\geqslant 1$ are given real numbers, $\alpha,\beta$ are real numbers such that $0<\alpha,\beta\leqslant 1$ . Define

T(N_{1},N_{2},\alpha,\beta)=\sum_{\begin{subarray}{c}n_{1}\sim N_{1},n_{2}\sim N_{2}\\ n_{1}\neq n_{2}\end{subarray}}\frac{1}{|{n_{1}}^{\alpha}-{n_{2}}^{\alpha}|^{\beta}}.

Then we have

T(N_{1},N_{2},\alpha,\beta)\ll(N_{1}N_{2})^{1-\alpha\beta/2}\log N_{1}\log N_{2}.

Proof.

We divide the sum into

\displaystyle\sum_{\begin{subarray}{c}n_{1}\sim N_{1}\\ n_{2}\sim N_{2}\end{subarray}}(|{n_{1}}^{\alpha}-{n_{2}}^{\alpha}|)^{-\beta}={\sum}_{1}+{\sum}_{2},

where

	$\displaystyle{\sum}_{1}$	$\displaystyle=\sum_{\begin{subarray}{c}n_{1}\sim N_{1},n_{2}\sim N_{2}\\ \|{n_{1}}^{\alpha}-{n_{2}}^{\alpha}\|\leqslant\frac{(n_{1}n_{2})^{\alpha/2}}{100}\end{subarray}}(\|{n_{1}}^{\alpha}-{n_{2}}^{\alpha}\|)^{-\beta},$
	$\displaystyle{\sum}_{2}$	$\displaystyle=\sum_{\begin{subarray}{c}n_{1}\sim N_{1},n_{2}\sim N_{2}\\ \|{n_{1}}^{\alpha}-{n_{2}}^{\alpha}\|\geqslant\frac{(n_{1}n_{2})^{\alpha/2}}{100}\end{subarray}}(\|{n_{1}}^{\alpha}-{n_{2}}^{\alpha}\|)^{-\beta}.$

Since $|{n_{1}}^{\alpha}-{n_{2}}^{\alpha}|\leqslant\frac{(n_{1}n_{2})^{\alpha/2}}{100}$ , we have $n_{1}\asymp n_{2}$ in ${\sum}_{1}$ . By the Lagrange mean value Theorem we have

	$\displaystyle{\sum}_{1}$	$\displaystyle\ll\sum_{\begin{subarray}{c}n_{1}\sim N_{1},n_{2}\sim N_{2}\\ n_{1}\asymp n_{2},n_{1}\neq n_{2}\end{subarray}}\frac{1}{{n_{1}}^{(\alpha-1)\beta}\|n_{1}-n_{2}\|^{\beta}}$
		$\displaystyle\ll\sum_{n_{1}\sim N_{1}}\frac{1}{{n_{1}}^{(\alpha-1)\beta}}\sum_{\begin{subarray}{c}n_{2}\sim N_{2}\\ n_{1}\asymp n_{2}\\ n_{1}\neq n_{2}\end{subarray}}\frac{1}{\|n_{1}-n_{2}\|^{\beta}},$

let $|n_{1}-n_{2}|=t$ in the latter sum, we have

	$\displaystyle{\sum}_{1}$	$\displaystyle\ll\sum_{n_{1}\sim N_{1}}\frac{1}{{n_{1}}^{(\alpha-1)\beta}}\sum_{t\leqslant n_{1}}t^{-\beta}$
		$\displaystyle\ll\sum_{n_{1}\sim N_{1}}\frac{1}{{n_{1}}^{\alpha\beta-1}}\log{N_{1}}$
		$\displaystyle\ll{N_{1}}^{2-\alpha\beta}\log{N_{1}}.$

Similarly we have ${\sum}_{1}\ll{N_{2}}^{2-\alpha\beta}\log{N_{2}}$ , thus

{\sum}_{1}\ll(N_{1}N_{2})^{1-\alpha\beta/2}\log{N_{1}}\log{N_{2}}.

(2.3)

For ${\sum}_{2}$ ,

{\sum}_{2}\ll\sum_{n_{1}\sim N_{1},n_{2}\sim N_{2}}\frac{1}{(n_{1}n_{2})^{\alpha\beta/2}}\ll(N_{1}N_{2})^{1-\alpha\beta/2}.

(2.4)

Thus we finish the proof by (2.3) and (2.4). ∎

We state a multiple exponential sums of the following form:

\mathop{\sum_{m_{1}\sim M_{1}}\sum_{m_{2}\sim M_{2}}}_{\begin{subarray}{c}m_{1}\neq m_{2}\end{subarray}}d(m_{1})d(m_{2})\mathop{\sum_{h\sim H}\sum_{\ell\sim L}}_{\begin{subarray}{c}HL\asymp x\end{subarray}}\xi_{h}\eta_{\ell}\mathbf{e}\left(2(\sqrt{m_{1}}-\sqrt{m_{2}})h^{1/2}{\ell}^{1/2}\right),

(2.5)

with $\xi_{h},\eta_{\ell}\in\mathbb{C},|\xi_{h}|\ll x^{\varepsilon},|\eta_{\ell}|\ll x^{\varepsilon}$ and $M_{1},M_{2},H,L,x$ are given real numbers, such that $1\ll M_{1},M_{2}\ll x$ . It is called a ”Type I” sum, denoted by $S_{I}$ , if $\eta_{\ell}=1$ or $\eta_{\ell}=\log\ell$ ; otherwise it is called a ”Type II” sum, denoted by $S_{II}$ .

Lemma 2.12.

Suppose that $\xi_{h}\ll 1$ , $\eta_{\ell}=1$ or $\eta_{\ell}=\log\ell$ , $HL\asymp x$ . Then if $H\ll x^{1/4}$ , there holds

x^{-\varepsilon}S_{I}\ll x^{1/2}{M_{1}}^{5/4}M_{2}+x^{3/4}(M_{1}M_{2})^{7/8}.

Proof.

Set $f(\ell)=2({m_{1}}^{1/2}-{m_{2}}^{1/2})(h\ell)^{1/2}$ . It is easy to see that

f^{{}^{\prime\prime}}(\ell)=-\frac{1}{2}\ell^{-3/2}h^{1/2}(\sqrt{m_{1}}-\sqrt{m_{2}})\asymp|\sqrt{m_{1}}-\sqrt{m_{2}}|L^{-3/2}H^{1/2}.

If $H\ll x^{1/4}$ , then by Lemma 2.9, we deduce that

	$\displaystyle x^{-\varepsilon}S_{I}$	$\displaystyle\ll\mathop{\sum_{m_{1}\sim M_{1}}\sum_{m_{2}\sim M_{2}}}_{\begin{subarray}{c}m_{1}\neq m_{2}\end{subarray}}\sum_{h\sim H}\left\|\sum_{\ell\sim L}\mathbf{e}(f(\ell))\right\|$
		$\displaystyle\ll\mathop{\sum_{m_{1}\sim M_{1}}\sum_{m_{2}\sim M_{2}}}_{\begin{subarray}{c}m_{1}\neq m_{2}\end{subarray}}\sum_{h\sim H}\left(L(L^{-3/2}H^{1/2}\|\sqrt{m_{1}}-\sqrt{m_{2}}\|)^{1/2}+(L^{-3/2}H^{1/2}\|\sqrt{m_{1}}-\sqrt{m_{2}}\|)^{-1/2}\right)$
		$\displaystyle\ll x^{1/4}H{M_{1}}^{5/4}M_{2}+x^{3/4}\mathop{\sum_{m_{1}\sim M_{1}}\sum_{m_{2}\sim M_{2}}}_{\begin{subarray}{c}m_{1}\neq m_{2}\end{subarray}}\|\sqrt{m_{1}}-\sqrt{m_{2}}\|^{-1/2},$

taking $n_{1}=m_{1},n_{2}=m_{2}$ , $\alpha=\beta=1/2$ in Lemma 2.11 we have

	$\displaystyle x^{-\varepsilon}S_{I}$	$\displaystyle\ll x^{1/4}H{M_{1}}^{5/4}M_{2}+x^{3/4}(M_{1}M_{2})^{7/8}$
		$\displaystyle\ll x^{1/2}{M_{1}}^{5/4}M_{2}+x^{3/4}(M_{1}M_{2})^{7/8}.$

∎

In order to separate the dependence from the range of summation we appeal to the following formula

Lemma 2.13.

Let $0<V\leqslant U<\nu U<\lambda V$ and let $a_{v}$ be complex numbers with $|a_{v}|\leqslant 1$ . We then have

\sum_{U<u<\nu U}a_{u}=\frac{1}{2\pi}\int_{-V}^{V}\left(\sum_{V<v<\lambda V}a_{v}v^{-it}\right)U^{it}(\nu^{it}-1)t^{-1}dt+O(\log(2+V)),

where the constant implied in $O$ depends on $\lambda$ only.

Proof.

See Lemma 6 of Fouvry and Iwaniec[2]. ∎

Lemma 2.14.

Suppose $|\xi_{h}|\ll x^{\varepsilon},|\eta_{\ell}|\ll x^{\varepsilon}$ with $h\sim H$ , $\ell\sim L$ , $HL\asymp x$ . Then for $x^{1/4}\ll H\ll{x^{1/2}}$ , there holds

	$\displaystyle x^{-\varepsilon}S_{II}$	$\displaystyle\ll x^{3/4}{M_{1}}^{9/8}{M_{2}}^{7/8}+x^{7/8}M_{1}{M_{2}}^{3/4}+x^{13/16}{M_{1}}^{19/16}{M_{2}}^{3/4}+x^{15/16}(M_{1}M_{2})^{3/4}$
		$\displaystyle+x^{1/2}M_{1}M_{2}.$

Proof.

Applying Lemma 2.13 we obtain

S_{II}\ll|{S_{II}}^{*}|+M_{1}M_{2}H\log x,

(2.6)

where

{S_{II}}^{*}=\mathop{\sum_{m_{1}\sim M_{1}}\sum_{m_{2}\sim M_{2}}}_{\begin{subarray}{c}m_{1}\neq m_{2}\end{subarray}}d(m_{1})d(m_{2})\sum_{h\sim H}{\xi_{h}}^{*}\sum_{\ell\sim L}{\eta_{\ell}}^{*}\mathbf{e}\left(2(\sqrt{m_{1}}-\sqrt{m_{2}})h^{1/2}{\ell}^{1/2}\right)

with $|{\xi_{h}}^{*}|,|{\eta_{\ell}}^{*}|\ll x^{\varepsilon}$ .

Applying Lemma 2.3 with $\mathscr{X}=2({m_{1}}^{1/2}-{m_{2}}^{1/2})h^{1/2}$ , $\mathscr{Y}=\ell^{1/2}$ , we obtain

{S_{II}}^{*}\ll{M_{1}}^{1/4}x^{1/4}(S_{a}S_{b})^{1/2},

(2.7)

where

	$\displaystyle S_{a}=$	$\displaystyle\sum_{\|({m_{1}}^{1/2}-{m_{2}}^{1/2}){h}^{1/2}-({\tilde{m_{1}}}^{1/2}-{\tilde{m_{2}}}^{1/2}){\tilde{h}}^{1/2}\|\leqslant L^{{-1/2}}}d(m_{1})d(m_{2})d(\tilde{m_{1}})d(\tilde{m_{2}}){\xi_{h}}^{}{\xi_{\tilde{h}}}^{},$
	$\displaystyle S_{b}=$	$\displaystyle\sum_{\|\ell^{1/2}-{\tilde{\ell}}^{1/2}\|\leqslant{M_{1}}^{-1/2}H^{-1/2}}{\eta_{\ell}}^{}{\eta_{\tilde{\ell}}}^{}.$

By (3.2) with $\Delta=L^{-1/2}$ , we easily obtain

	$\displaystyle x^{-\varepsilon}S_{a}$	$\displaystyle\ll\mathscr{B}(M_{1},M_{2},H,L^{-1/2})$		(2.8)
		$\displaystyle\ll(M_{1}M_{2})^{3/2}H^{3/2}+{M_{1}}^{15/8}{M_{2}}^{3/2}x^{-1/8}H^{3/2}+(M_{1}M_{2})^{7/4}x^{-1/2}H^{2}.$		(2.8)

For $S_{b}$ , using Lemma 2.6 we have

	$\displaystyle x^{-\varepsilon}S_{b}$	$\displaystyle\ll\sum_{\|\ell^{1/2}-{\tilde{\ell}}^{1/2}\|\leqslant{M_{1}}^{-1/2}H^{-1/2}}1$		(2.9)
		$\displaystyle\ll xH^{-1}+{M_{1}}^{-1/2}H^{-2}x^{3/2}.$		(2.9)

Thus combining (2.6), (2.7), (2.8), (2.9) and the condition $x^{1/4}\ll H\ll x^{1/2}$ we conclude that

$\displaystyle x^{-\varepsilon}{S_{II}}^{*}$	$\displaystyle\ll x^{1/2}H^{1/2}{M_{1}}^{9/8}{M_{2}}^{7/8}+x^{3/4}H^{1/4}M_{1}{M_{2}}^{3/4}+H^{1/4}x^{11/16}{M_{1}}^{19/16}{M_{2}}^{3/4}$	(2.10)
	$\displaystyle+x^{3/4}(M_{1}M_{2})^{7/8}+xH^{-1/4}(M_{1}M_{2})^{3/4}+x^{15/16}H^{-1/4}{M_{1}}^{15/16}{M_{2}}^{3/4}+x^{1/2}M_{1}M_{2}$
	$\displaystyle\ll x^{3/4}{M_{1}}^{9/8}{M_{2}}^{7/8}+x^{7/8}M_{1}{M_{2}}^{3/4}+x^{13/16}{M_{1}}^{19/16}{M_{2}}^{3/4}+x^{15/16}(M_{1}M_{2})^{3/4}$
	$\displaystyle+x^{1/2}M_{1}M_{2}.$

∎

3. The spacing problem

Let $M_{1},M_{2}\geqslant 1$ , $H\geqslant 1$ . In this section $\Delta$ is a small real positive number. In this section we investigate the distributions of real numbers of type

t(h,m_{1},m_{2})=({m_{1}}^{\alpha}-{m_{2}}^{\alpha})h^{\beta}

with $0<\alpha,\beta<1$ , $m_{1}\sim M_{1}$ , $m_{2}\sim M_{2}$ , $m_{1}\neq m_{2}$ , $h\sim H$ .

Let $\mathscr{B}(M_{1},M_{2},H,\Delta)$ denote the number of sixturplets $(m_{1},\tilde{m_{1}},m_{2},\tilde{m_{2}},h,\tilde{h})$ with $m_{1},\tilde{m_{1}}\sim M_{1}$ , $m_{2},\tilde{m_{2}}\sim M_{2}$ , $h,\tilde{h}\sim H$ satisfying

|t(h,m_{1},m_{2})-t(\tilde{h},\tilde{m_{1}},\tilde{m_{2}})|\leqslant\Delta

(3.1)

Our aim is to prove the following:

Proposition 3.1.

We have

		$\displaystyle\mathscr{B}(M_{1},M_{2},H,\Delta)$		(3.2)
		$\displaystyle\ll(M_{1}M_{2})^{3/2+\varepsilon}H^{3/2}+{M_{1}}^{2-\alpha/4+\varepsilon}{M_{2}}^{3/2+\varepsilon}\Delta^{1/4}H^{3/2-\beta/4}+(M_{1}M_{2})^{2-\alpha/2+\varepsilon}H^{2-\beta}\Delta.$		(3.2)

Proof.

Suppose $m_{1},m_{2},\tilde{m_{1}},\tilde{m_{2}}$ are fixed, without loss of generality we assume that $m_{1}>m_{2}$ and $\tilde{m_{1}}>\tilde{m_{2}}$ . We denote the number of solutions of

\left|\left(\frac{{m_{1}}^{\alpha}-{m_{2}}^{\alpha}}{{\tilde{m_{1}}}^{\alpha}-{\tilde{m_{2}}}^{\alpha}}\right)-\left(\frac{h}{\tilde{h}}\right)^{\beta}\right|\leq\frac{\Delta}{|{\tilde{m_{1}}}^{\alpha}-{\tilde{m_{2}}}^{\alpha}|H^{\beta}}

(3.3)

by $\mathscr{B}(m_{1},m_{2},\tilde{m_{1}},\tilde{m_{2}},H,\Delta)$ . Let $\mathscr{B}(m_{1},m_{2},\tilde{m_{1}},\tilde{m_{2}},H,\Delta,\mu)$ be the number of solutions of $h,\tilde{h}$ to (3.3) which additionally satisfying $\gcd(h,\tilde{h})=\mu$ .

We divide the solutions of $h$ , $\tilde{h}$ into classes each one having fixed values $\gcd(h,\tilde{h})=\mu$ , the points $\left(\frac{h}{\tilde{h}}\right)$ are spaced by $c(\beta)H^{2}\mu^{-2}$ , where $c(\beta)$ is a constant. Then

\mathscr{B}(m_{1},m_{2},\tilde{m_{1}},\tilde{m_{2}},H,\Delta,\mu)\ll 1+\Delta(|{\tilde{m_{1}}}^{\alpha}-{\tilde{m_{2}}}^{\alpha}|H^{\beta})^{-1}H^{2}\mu^{-2}

Let $\mathscr{B}(M_{1},M_{2},H,\Delta,\mu)$ be the number of solutions to (3.1) which additionally satisfying $\gcd(h,\tilde{h})=\mu$ . Summing over $m_{1}$ , $m_{2}$ , $\tilde{m_{1}}$ , $\tilde{m_{2}}$ we have

	$\displaystyle\mathscr{B}(M_{1},M_{2},H,\Delta,\mu)$	$\displaystyle\ll\sum_{\begin{subarray}{c}m_{1},\tilde{m_{1}}\sim M_{1}\\ m_{2},\tilde{m_{2}}\sim M_{2}\end{subarray}}\mathscr{B}(m_{1},m_{2},\tilde{m_{1}},\tilde{m_{2}},H,\Delta,\mu)$
		$\displaystyle\ll\sum_{\begin{subarray}{c}m_{1},\tilde{m_{1}}\sim M_{1}\\ m_{2},\tilde{m_{2}}\sim M_{2}\end{subarray}}(1+\Delta(\|{\tilde{m_{1}}}^{\alpha}-{\tilde{m_{2}}}^{\alpha}\|)^{-1}H^{2-\beta}\mu^{-2})$
		$\displaystyle\ll{M_{1}}^{2}{M_{2}}^{2}+\Delta M_{1}M_{2}H^{2-\beta}\mu^{-2}\sum_{\begin{subarray}{c}\tilde{m_{1}}\sim M_{1}\\ \tilde{m_{2}}\sim M_{2}\end{subarray}}(\|{\tilde{m_{1}}}^{\alpha}-{\tilde{m_{2}}}^{\alpha}\|)^{-1}.$

Using Lemma 2.11 we conclude that

\displaystyle\mathscr{B}(M_{1},M_{2},H,\Delta,\mu)\ll{M_{1}}^{2}{M_{2}}^{2}+\Delta(M_{1}M_{2})^{2-\alpha/2}H^{2-\beta}\mu^{-2}\log{M_{1}}\log{M_{2}}.

Summing over $\mu$ we have

\displaystyle\mathscr{B}(M_{1},M_{2},H,\Delta)\ll H{M_{1}}^{2}{M_{2}}^{2}+\Delta(M_{1}M_{2})^{2-\alpha/2}H^{2-\beta}\log{M_{1}}\log{M_{2}}.

(3.4)

On the other hand, for fixed $h$ , $\tilde{h}$ , we denote the number of solutions of $m_{1},\tilde{m_{1}},m_{2},\tilde{m_{2}}$ to

\left|\left(\frac{h}{\tilde{h}}\right)^{\beta}({m_{1}}^{\alpha}-{m_{2}}^{\alpha})-{\tilde{m_{1}}}^{\alpha}+{\tilde{m_{2}}}^{\alpha}\right|\leq\frac{\Delta}{H^{\beta}}

(3.5)

by $\mathscr{B}(M_{1},M_{2},h,\tilde{h},H,\Delta)$ . Then we can obtain that $\mathscr{B}(M_{1},M_{2},H,\Delta)$ is bounded by

\sum_{h,\tilde{h}\sim H}\mathscr{B}(M_{1},M_{2},h,\tilde{h},H,\Delta).

Applying Lemma 2.4 to the sequences $\{a_{r}\}=\{(h/\tilde{h})^{\beta}{m_{1}}^{\alpha}-{\tilde{m_{1}}}^{\alpha}\}$ and $\{b_{s}\}=\{(h/\tilde{h})^{\beta}{m_{2}}^{\alpha}-{\tilde{m_{2}}}^{\alpha}\}$ , we get

		$\displaystyle\mathscr{B}(M_{1},M_{2},h,\tilde{h},H,\Delta)$
		$\displaystyle\qquad\leqslant 3\left(\#\left\{(m_{1},\tilde{m_{1}},{m_{1}}^{\prime},{\tilde{m_{1}}^{\prime}}):\left\|\left(\frac{h}{\tilde{h}}\right)^{\beta}{m_{1}}^{\alpha}-{\tilde{m_{1}}}^{\alpha}-\left(\frac{h}{\tilde{h}}\right)^{\beta}{{m_{1}}^{\prime}}^{\alpha}+{\tilde{m_{1}}{{}^{\prime}}}^{\alpha}\right\|\leqslant\frac{\Delta}{H^{\beta}}\right\}\right)^{1/2}$
		$\displaystyle\qquad\times\left(\#\left\{(m_{2},\tilde{m_{2}},{m_{2}}^{\prime},{\tilde{m_{2}}^{\prime}}):\left\|\left(\frac{h}{\tilde{h}}\right)^{\beta}{m_{2}}^{\alpha}-{\tilde{m_{2}}}^{\alpha}-\left(\frac{h}{\tilde{h}}\right)^{\beta}{{m_{2}}^{\prime}}^{\alpha}+{\tilde{m_{2}}{{}^{\prime}}}^{\alpha}\right\|\leqslant\frac{\Delta}{H^{\beta}}\right\}\right)^{1/2}.$

Applying Lemma 2.4 again to the sequences $\{a_{r_{1}}\}=\{(h/\tilde{h})^{\beta}({m_{1}}^{\alpha}-{{m_{1}}{{}^{\prime}}^{\alpha}})\}$ , $\{b_{s_{1}}\}=\{{\tilde{m_{1}}}^{\alpha}-{\tilde{m_{1}}{{}^{\prime}}}^{\alpha}\}$ , and $\{a_{r_{2}}\}=\{(h/\tilde{h})^{\beta}({m_{2}}^{\alpha}-{{m_{2}}{{}^{\prime}}^{\alpha}})\}$ , $\{b_{s_{2}}\}=\{{\tilde{m_{2}}}^{\alpha}-{\tilde{m_{2}}{{}^{\prime}}}^{\alpha}\}$ , respectively, we get

\displaystyle\mathscr{B}(M_{1},M_{2},h,\tilde{h},H,\Delta)\ll\mathcal{N}\left(M_{1},\frac{\Delta}{H^{\beta}{M_{1}}^{\alpha}}\right)^{1/2}\mathcal{N}\left(M_{2},\frac{\Delta}{H^{\beta}{M_{2}}^{\alpha}}\right)^{1/2},

where $\mathcal{N}$ is defined in Lemma 2.5. Summing over $h$ and $\tilde{h}$ we have

	$\displaystyle\mathscr{B}(M_{1},M_{2},H,\Delta)$	$\displaystyle\ll\sum_{h,\tilde{h}\sim H}\mathscr{B}(M_{1},M_{2},h,\tilde{h},H,\Delta)$		(3.6)
		$\displaystyle\ll(H{M_{1}}^{1+\varepsilon}+{M_{1}}^{2-\alpha/2+\varepsilon}\Delta^{1/2}H^{1-\beta/2})(H{M_{2}}^{1+\varepsilon}+{M_{2}}^{2-\alpha/2+\varepsilon}\Delta^{1/2}H^{1-\beta/2})$		(3.6)

Above all, combining (3.4) and (3.6) we get

		$\displaystyle\mathscr{B}(M_{1},M_{2},H,\Delta)$
	$\displaystyle\ll$	$\displaystyle\min({M_{1}}^{2}{M_{2}}^{2}+\Delta(M_{1}M_{2})^{2-\alpha/2}H^{2-\beta}\log{M_{1}}\log{M_{2}},$
		$\displaystyle(H{M_{1}}^{1+\varepsilon}+{M_{1}}^{2-\alpha/2+\varepsilon}\Delta^{1/2}H^{1-\beta/2})(H{M_{2}}^{1+\varepsilon}+{M_{2}}^{2-\alpha/2+\varepsilon}\Delta^{1/2}H^{1-\beta/2}))$
	$\displaystyle\ll$	$\displaystyle\min(H{M_{1}}^{2}{M_{2}}^{2},H^{2}(M_{1}M_{2})^{1+\varepsilon}+{M_{1}}^{2-\alpha/2+\varepsilon}{M_{2}}^{1+\varepsilon}\Delta^{1/2}H^{2-\beta/2})+\Delta(M_{1}M_{2})^{2-\alpha/2+\varepsilon}H^{2-\beta}$
	$\displaystyle\ll$	$\displaystyle(M_{1}M_{2})^{3/2+\varepsilon}H^{3/2}+{M_{1}}^{2-\alpha/4+\varepsilon}{M_{2}}^{3/2+\varepsilon}\Delta^{1/4}H^{3/2-\beta/4}+(M_{1}M_{2})^{2-\alpha/2+\varepsilon}H^{2-\beta}\Delta,$

where we use $\min(a,b)\ll(ab)^{1/2}$ . ∎

4. Proof of the Theorem

We are first going to estimate $\sum_{x<p\leqslant 2x}\Delta^{2}(p)$ . Using notations in Lemma 2.1, let $M$ be a real number such that $1\ll M\ll x$ , one has

\displaystyle\sum_{x<p\leqslant 2x}\Delta^{2}(p)=\sum_{x<p\leqslant 2x}(\delta_{1}(p,M)^{2}+2\delta_{1}(p,M)\delta_{2}(p,M)+\delta_{2}(p,M)^{2}),

(4.1)

denoted by $S_{11},S_{12},S_{22}$ , respectively.

For $S_{22}$ , one has

	$\displaystyle S_{22}$	$\displaystyle=\sum_{x<p\leqslant 2x}\int_{p-1}^{p}\delta_{2}(p,M)^{2}dt$
		$\displaystyle=\sum_{x<p\leqslant 2x}\left[\int_{p-1}^{p}\delta_{2}(t,M)^{2}dt-\int_{p-1}^{p}(\delta_{2}(t,M)^{2}-\delta_{2}(p,M)^{2})dt\right]$
		$\displaystyle\leqslant\int_{x}^{2x}\delta_{2}(t,M)^{2}dt+\sum_{x<p\leqslant 2x}\int_{p-1}^{p}(\delta_{2}(t,M)^{2}-\delta_{2}(p,M)^{2})dt$
		$\displaystyle\ll\frac{x^{3/2}\log^{3}x}{{M}^{1/2}}+x\log^{4}x+\sum_{x<p\leqslant 2x}\int_{p-1}^{p}(\delta_{2}(t,M)^{2}-\delta_{2}(p,M)^{2})dt.$

The latter term can be transformed into $E_{1}-E_{2}$ , where

	$\displaystyle E_{1}$	$\displaystyle=\sum_{x<p\leqslant 2x}\int_{p-1}^{p}(\delta_{2}(t,M)+\delta_{2}(p,M))(\Delta(t)-\Delta(p))dt,$
	$\displaystyle E_{2}$	$\displaystyle=\sum_{x<p\leqslant 2x}\int_{p-1}^{p}(\delta_{2}(t,M)+\delta_{2}(p,M))(\delta_{1}(t,M)-\delta_{1}(p,M))dt.$

For $E_{2}$ , by the expression of $\delta_{1}$ and Lagrange’s mean value theorem we obtain $\delta_{1}(t,M)-\delta_{1}(p,M)\ll t^{-1/4}{M}^{3/4}$ , and by Lemma 2.1 we have $\delta_{2}(t,M)+\delta_{2}(p,M)\ll t^{1/2+\varepsilon}{M}^{-1/2}$ , thus

E_{2}\ll\int_{x}^{2x}t^{1/4+\varepsilon}{M}^{1/4}dt\ll x^{5/4+\varepsilon}{M}^{1/4}.

For $E_{1}$ , one has

	$\displaystyle E_{1}$	$\displaystyle\ll\int_{x}^{2x}t^{1/2}{M}^{-1/2}\max_{0<v\leqslant 1}\|\Delta(t)-\Delta(t+v)\|dt$
		$\displaystyle\ll\frac{x^{1/2+\varepsilon}}{{M}^{1/2}}\int_{x}^{2x}\max_{0<v\leqslant 1}\|\Delta(t)-\Delta(t+v)\|dt$
		$\displaystyle\ll x^{3/2+\varepsilon}{M}^{-1/2},$

where we use Lemma 2.10. Thus we conclude that

S_{22}\ll x^{3/2+\varepsilon}M^{-1/2}+x^{5/4+\varepsilon}{M}^{1/4}.

(4.2)

Next we turn to evaluate $S_{11}$ . Using $\cos{\alpha}\cos{\beta}=\frac{1}{2}(\cos(\alpha-\beta)+\cos(\alpha+\beta))$ we have

	$\displaystyle\sum_{x<p\leqslant 2x}\delta_{1}(p,M)^{2}$	(4.3)
$\displaystyle=$	$\displaystyle\sum_{x<p\leqslant 2x}\frac{p^{1/2}}{2\pi^{2}}\sum_{m_{1},m_{2}\leqslant M}\frac{d(m_{1})d(m_{2})}{(m_{1}m_{2})^{3/4}}\cos\left(4\pi\sqrt{m_{1}p}-\frac{\pi}{4}\right)\cos\left(4\pi\sqrt{m_{2}p}-\frac{\pi}{4}\right)$
$\displaystyle=$	$\displaystyle\frac{1}{4\pi^{2}}\sum_{x<p\leqslant 2x}p^{1/2}\sum_{m_{1},m_{2}\leqslant M}\frac{d(m_{1})d(m_{2})}{(m_{1}m_{2})^{3/4}}\left(\cos\left(4\pi(\sqrt{m_{1}}-\sqrt{m_{2}})\sqrt{p}\right)+\sin\left(4\pi(\sqrt{m_{1}}+\sqrt{m_{2}})\sqrt{p}\right)\right)$
$\displaystyle:=$	$\displaystyle\frac{1}{4\pi^{2}}\sum_{x<p\leqslant 2x}p^{1/2}\sum_{m\leqslant M}\frac{d^{2}(m)}{m^{3/2}}+R_{1}+R_{2},$

where

	$\displaystyle R_{1}$	$\displaystyle=\frac{1}{4\pi^{2}}\sum_{\begin{subarray}{c}m_{1},m_{2}\leqslant M\\ m_{1}\neq m_{2}\end{subarray}}\frac{d(m_{1})d(m_{2})}{(m_{1}m_{2})^{3/4}}\sum_{x<p\leqslant 2x}p^{1/2}\cos\left(4\pi(\sqrt{m_{1}}-\sqrt{m_{2}})\sqrt{p}\right),$
	$\displaystyle R_{2}$	$\displaystyle=\frac{1}{4\pi^{2}}\sum_{m_{1},m_{2}\leqslant M}\frac{d(m_{1})d(m_{2})}{(m_{1}m_{2})^{3/4}}\sum_{x<p\leqslant 2x}p^{1/2}\sin\left(4\pi(\sqrt{m_{1}}+\sqrt{m_{2}})\sqrt{p}\right).$

4.1. Estimate of $R_{1}$

One can see $R_{1}$ can be written as linear combination of $O(\log^{2}{M})$ sums of the form

		$\displaystyle\frac{1}{4\pi^{2}}\sum_{\begin{subarray}{c}m_{1}\sim M_{1},m_{2}\sim M_{2}\\ m_{1}\neq m_{2}\end{subarray}}\frac{d(m_{1})d(m_{2})}{(m_{1}m_{2})^{3/4}}\sum_{x<p\leqslant 2x}p^{1/2}\cos\left(4\pi(\sqrt{m_{1}}-\sqrt{m_{2}})\sqrt{p}\right)$
	$\displaystyle=$	$\displaystyle\frac{1}{8\pi^{2}}\sum_{\begin{subarray}{c}m_{1}\sim M_{1},m_{2}\sim M_{2}\\ m_{1}\neq m_{2}\end{subarray}}\frac{d(m_{1})d(m_{2})}{(m_{1}m_{2})^{3/4}}\sum_{x<p\leqslant 2x}p^{1/2}(\mathbf{e}(2(\sqrt{m_{1}}-\sqrt{m_{2}})\sqrt{p})+\mathbf{e}(2(\sqrt{m_{2}}-\sqrt{m_{1}})\sqrt{p})),$

by using $\cos 2\pi\alpha=(\mathbf{e}(\alpha)+\mathbf{e}(-\alpha))/2$ , and where $1\leqslant M_{1},M_{2}\leqslant M$ . Since $m_{1},m_{2}$ are symmetric in the above sum, we are just going to estimate

\mathcal{S}_{1}=\frac{1}{8\pi^{2}}\sum_{\begin{subarray}{c}m_{1}\sim M_{1},m_{2}\sim M_{2}\\ m_{1}\neq m_{2}\end{subarray}}\frac{d(m_{1})d(m_{2})}{(m_{1}m_{2})^{3/4}}\sum_{x<p\leqslant 2x}p^{1/2}\>\mathbf{e}(2(\sqrt{m_{1}}-\sqrt{m_{2}})\sqrt{p}).

Trivially we have

\mathcal{S}_{1}\ll\frac{x^{1/2}}{(M_{1}M_{2})^{3/4}}|{\mathcal{S}_{1}}^{*}|,

(4.4)

where

{\mathcal{S}_{1}}^{*}=\sum_{\begin{subarray}{c}m_{1}\sim M_{1},m_{2}\sim M_{2}\\ m_{1}\neq m_{2}\end{subarray}}d(m_{1})d(m_{2})\sum_{x<p\leqslant 2x}\>\mathbf{e}(2(\sqrt{m_{1}}-\sqrt{m_{2}})\sqrt{p}).

It follows from partial summation that

	$\displaystyle{\mathcal{S}_{1}}^{*}$	$\displaystyle=\sum_{\begin{subarray}{c}m_{1}\sim M_{1},m_{2}\sim M_{2}\\ m_{1}\neq m_{2}\end{subarray}}d(m_{1})d(m_{2})\int_{x}^{2x}\frac{d\mathcal{A}(u)}{\log u}$		(4.5)
		$\displaystyle=\sum_{\begin{subarray}{c}m_{1}\sim M_{1},m_{2}\sim M_{2}\\ m_{1}\neq m_{2}\end{subarray}}d(m_{1})d(m_{2})\left(\frac{\mathcal{A}(u)}{\log u}\bigg{\|}_{x}^{2x}-\int_{x}^{2x}\frac{\mathcal{A}(u)}{u\log^{2}u}du\right),$		(4.5)

where

\mathcal{A}(u)=\sum_{p\leqslant u}\log p\>\mathbf{e}(2(\sqrt{m_{1}}-\sqrt{m_{2}})\sqrt{p}).

Moreover, we easily obtain

\mathcal{A}(u)=\sum_{n\leqslant u}\Lambda(n)\>\mathbf{e}(2(\sqrt{m_{1}}-\sqrt{m_{2}})\sqrt{n})+O(u^{1/2}).

By a splitting argument we only need to give the upper bound estimate of the following sum

\sum_{\begin{subarray}{c}m_{1}\sim M_{1},m_{2}\sim M_{2}\\ m_{1}\neq m_{2}\end{subarray}}d(m_{1})d(m_{2})\sum_{n\sim x}\Lambda(n)\>\mathbf{e}(2(\sqrt{m_{1}}-\sqrt{m_{2}})\sqrt{n}).

(4.6)

After using Heath-Brown’s identity, i.e. Lemma 2.2 with $k=3$ , one can see that (4.6) can be written as linear combination of $O(\log^{6}x)$ sums, each of which is of the form

	$\displaystyle\mathcal{T}^{*}:=$	$\displaystyle\sum_{\begin{subarray}{c}m_{1}\sim M_{1},m_{2}\sim M_{2}\\ m_{1}\neq m_{2}\end{subarray}}d(m_{1})d(m_{2}){\sum_{n_{1}\sim N_{1}}}\cdots{\sum_{n_{6}\sim N_{6}}}(\log n_{1})\mu(n_{4})\mu(n_{5})\mu(n_{6})$		(4.7)
		$\displaystyle\times\mathbf{e}(2(\sqrt{m_{1}}-\sqrt{m_{2}})\sqrt{n_{1}\cdots n_{6}}),$		(4.7)

where $N_{1}\cdots N_{6}\asymp x$ ; $2N_{i}\leqslant(2x)^{1/3}$ , $i=4,5,6$ and some $n_{i}$ may only take value $1$ . Therefore, it is sufficient for us to estimate for each $\mathcal{T}^{*}$ defined as in (4.7). Next, we will consider four cases.

Case 1

If there exists an $N_{j}$ such that $N_{j}\geqslant x^{3/4}$ , then we must have $j\leqslant 3$ for the fact that

$N_{j}\ll x^{1/3}$ with $j=4,5,6$ . Let

h=\prod_{\begin{subarray}{c}1\leqslant i\leqslant 6\\ i\neq j\end{subarray}}n_{i},\qquad\ell=n_{j},\qquad H=\prod_{\begin{subarray}{c}1\leqslant i\leqslant 6\\ i\neq j\end{subarray}}N_{i},\qquad L=N_{j}.

In this case, we can see that $\mathcal{T}^{*}$ is a sum of ”Type I” satisfying $H\ll x^{1/4}$ . By Lemma 2.12, we have

x^{-\varepsilon}\cdot\mathcal{T}^{*}\ll x^{1/2}{M_{1}}^{5/4}M_{2}+x^{3/4}(M_{1}M_{2})^{7/8}.

Case 2

If there exists an $N_{j}$ such that $x^{1/2}\leqslant N_{j}<x^{3/4}$ , then we take

h=\prod_{\begin{subarray}{c}1\leqslant i\leqslant 6\\ i\neq j\end{subarray}}n_{i},\qquad\ell=n_{j},\qquad H=\prod_{\begin{subarray}{c}1\leqslant i\leqslant 6\\ i\neq j\end{subarray}}N_{i},\qquad L=N_{j}.

Thus, $\mathcal{T}^{*}$ is a sum of ”Type II” satisfying $x^{1/4}\ll H\ll x^{1/2}$ . By Lemma 2.14, we have

	$\displaystyle x^{-\varepsilon}\cdot\mathcal{T}^{*}$	$\displaystyle\ll x^{3/4}{M_{1}}^{9/8}{M_{2}}^{7/8}+x^{7/8}M_{1}{M_{2}}^{3/4}+x^{13/16}{M_{1}}^{19/16}{M_{2}}^{3/4}+x^{15/16}(M_{1}M_{2})^{3/4}$
		$\displaystyle+x^{1/2}M_{1}M_{2}.$

Case 3

If there exists an $N_{j}$ such that $x^{1/4}\leqslant N_{j}<x^{1/2}$ , then we take

h=n_{j},\qquad\ell=\prod_{\begin{subarray}{c}1\leqslant i\leqslant 6\\ i\neq j\end{subarray}}n_{i},\qquad H=N_{j},\qquad L=\prod_{\begin{subarray}{c}1\leqslant i\leqslant 6\\ i\neq j\end{subarray}}N_{i}.

Thus, $\mathcal{T}^{*}$ is a sum of ”Type II” satisfying $x^{1/4}\ll H\ll x^{1/2}$ . By Lemma 2.14, we have

	$\displaystyle x^{-\varepsilon}\cdot\mathcal{T}^{*}$	$\displaystyle\ll x^{3/4}{M_{1}}^{9/8}{M_{2}}^{7/8}+x^{7/8}M_{1}{M_{2}}^{3/4}+x^{13/16}{M_{1}}^{19/16}{M_{2}}^{3/4}+x^{15/16}(M_{1}M_{2})^{3/4}$
		$\displaystyle+x^{1/2}M_{1}M_{2}.$

Case 4

If $N_{j}<x^{1/4}(j=1,2,3,4,5,6)$ , without loss of generality, we assume that

$N_{1}\geqslant N_{2}\geqslant\cdots\geqslant N_{6}$ . Let $r$ denote the natural number $j$ such that

N_{1}N_{2}\cdots N_{j-1}<x^{1/4},\qquad N_{1}N_{2}\cdots N_{j}\geqslant x^{1/4}.

Since $N_{1}<x^{1/4}$ and $N_{6}<x^{1/4}$ , then $2\leqslant r\leqslant 5$ . Thus we have

x^{1/4}\leqslant N_{1}N_{2}\cdots N_{r}=(N_{1}N_{2}\cdots N_{r-1})\cdot N_{r}<x^{1/4}\cdot x^{1/4}=x^{1/2}.

Let

h=\prod_{i=1}^{r}n_{i},\qquad\ell=\prod_{i=r+1}^{6}n_{i},\qquad H=\prod_{i=1}^{r}N_{i},\qquad L=\prod_{i=r+1}^{6}N_{i}.

At this time, $\mathcal{T}^{*}$ is a sum of ”Type II” satisfying $x^{1/4}\ll H\ll x^{1/2}$ . By Lemma 2.14, we have

	$\displaystyle x^{-\varepsilon}\cdot\mathcal{T}^{*}$	$\displaystyle\ll x^{3/4}{M_{1}}^{9/8}{M_{2}}^{7/8}+x^{7/8}M_{1}{M_{2}}^{3/4}+x^{13/16}{M_{1}}^{19/16}{M_{2}}^{3/4}+x^{15/16}(M_{1}M_{2})^{3/4}$
		$\displaystyle+x^{1/2}M_{1}M_{2}.$

Combining the above four cases, we derive that

	$\displaystyle x^{-\varepsilon}\cdot\mathcal{T}^{*}$	$\displaystyle\ll x^{3/4}{M_{1}}^{9/8}{M_{2}}^{7/8}+x^{7/8}M_{1}{M_{2}}^{3/4}+x^{13/16}{M_{1}}^{19/16}{M_{2}}^{3/4}+x^{15/16}(M_{1}M_{2})^{3/4}$
		$\displaystyle+x^{1/2}{M_{1}}^{5/4}M_{2},$

which combined with (4.4) and (4.5) yields

\displaystyle x^{-\varepsilon}\mathcal{S}_{1}

\displaystyle\ll x^{5/4}{M_{1}}^{3/8}{M_{2}}^{1/8}+x^{11/8}{M_{1}}^{1/4}+x^{21/16}{M_{1}}^{7/16}+x^{23/16}+x^{11/8}{M_{1}}^{3/16}+x{M_{1}}^{1/2}{M_{2}}^{1/4}.

Thus we obtain

\displaystyle x^{-\varepsilon}R_{1}\ll x^{5/4}{M_{1}}^{3/8}{M_{2}}^{1/8}+x^{11/8}{M_{1}}^{1/4}+x^{21/16}{M_{1}}^{7/16}+x^{23/16}+x^{11/8}{M_{1}}^{3/16}+x{M_{1}}^{1/2}{M_{2}}^{1/4}.

(4.8)

4.2. Estimate of $R_{2}$

By the same argument on $R_{1}$ , we derive that

R_{2}\ll\frac{x^{1/2}}{(M_{1}M_{2})^{3/4}}|{\mathcal{S}_{2}}^{*}|,

(4.9)

where

{\mathcal{S}_{2}}^{*}=\sum_{\begin{subarray}{c}m_{1}\sim M_{1},m_{2}\sim M_{2}\\ m_{1}\neq m_{2}\end{subarray}}d(m_{1})d(m_{2})\sum_{x<n\leqslant 2x}\Lambda(n)\>\mathbf{e}(2(\sqrt{m_{1}}+\sqrt{m_{2}})\sqrt{n}).

(4.10)

Using Lemma 2.7, Lemma 2.8 and the Cauchy-Schwarz’s inequality, one has

		$\displaystyle\sum_{x<n\leqslant 2x}\Lambda(n)\>\mathbf{e}(2(\sqrt{m_{1}}+\sqrt{m_{2}})\sqrt{n})$
		$\displaystyle\ll\left(\sum_{x<n\leqslant 2x}\Lambda^{2}(n)\right)^{1/2}\left(\sum_{x<n\leqslant 2x}\>\mathbf{e}(4(\sqrt{m_{1}}+\sqrt{m_{2}})\sqrt{n})\right)^{1/2}$
		$\displaystyle\ll x^{1/2+\varepsilon}\left(\int_{x}^{2x}\mathbf{e}(4(\sqrt{m_{1}}+\sqrt{m_{2}})\sqrt{t})dt\right)^{1/2}$
		$\displaystyle\ll x^{3/4+\varepsilon}(m_{1}m_{2})^{-1/8},$

which combined with (4.10) yields

	$\displaystyle R_{2}$	$\displaystyle\ll\frac{x^{5/4+\varepsilon}}{(M_{1}M_{2})^{3/4}}\sum_{\begin{subarray}{c}m_{1}\sim M_{1},m_{2}\sim M_{2}\\ m_{1}\neq m_{2}\end{subarray}}d(m_{1})d(m_{2})(m_{1}m_{2})^{-1/8}$		(4.11)
		$\displaystyle\ll x^{5/4+\varepsilon}(M_{1}M_{2})^{1/8}.$		(4.11)

Above all, since $M_{1},M_{2}$ run over $1\leqslant M_{1},M_{2}\leqslant M$ , $1\ll M\ll x$ , by (4.3), (4.8), (4.11), after eliminating lower order terms, we conclude that

$\displaystyle S_{11}$	$\displaystyle=\frac{1}{4\pi^{2}}\sum_{x<p\leqslant 2x}p^{1/2}\left(\sum_{m=1}^{\infty}\frac{d^{2}(m)}{m^{3/2}}-\sum_{m>M}\frac{d^{2}(m)}{m^{3/2}}\right)+O(x^{\varepsilon}(x^{5/4}{M_{1}}^{3/8}{M_{2}}^{1/8}+x^{11/8}{M_{1}}^{1/4}$	(4.12)
	$\displaystyle+x^{21/16}{M_{1}}^{7/16}+x^{23/16}+x^{11/8}{M_{1}}^{3/16}+x{M_{1}}^{1/2}{M_{2}}^{1/4}))$
	$\displaystyle=\frac{\mathcal{C}}{4\pi^{2}}\sum_{x<p\leqslant 2x}p^{1/2}+O(x^{\varepsilon}(x^{3/2}M^{-1/2}+x^{5/4}{M_{1}}^{3/8}{M_{2}}^{1/8}+x^{11/8}{M_{1}}^{1/4}+x^{21/16}{M_{1}}^{7/16}$
	$\displaystyle+x^{23/16}+x^{11/8}{M_{1}}^{3/16}+x{M_{1}}^{1/2}{M_{2}}^{1/4})),$
	$\displaystyle=\frac{\mathcal{C}}{4\pi^{2}}\sum_{x<p\leqslant 2x}p^{1/2}+O(x^{\varepsilon}(x^{3/2}M^{-1/2}+x^{5/4}M^{1/2}+x^{21/16}{M}^{7/16}+x^{23/16}+x^{11/8}{M}^{3/16})),$

where we use the partial summation and $\mathcal{C}$ denotes $\sum_{m=1}^{\infty}\frac{d^{2}(m)}{m^{3/2}}$ .

It remains to estimate $S_{12}$ , suppose $M\ll x^{3/7}$ , then by (4.2), (4.12) and the Cauchy-Schwarz’s inequality we have

S_{12}\ll(S_{11})^{1/2}(S_{22})^{1/2}\ll x^{3/2+\varepsilon}M^{-1/4}.

(4.13)

Since $1\ll M\ll x^{3/7}$ , combining (4.1), (4.2), (4.12), (4.13), taking $M=x^{1/4}$ , we obtain

\sum_{x<p\leqslant 2x}\Delta^{2}(p)=\frac{\mathcal{C}}{4\pi^{2}}\sum_{x<p\leqslant 2x}p^{1/2}+O(x^{23/16+\varepsilon}).

Thus replacing $x$ by $x/2$ , $x/2^{2}$ , and so on, and adding up all the results, we obtain

\displaystyle\sum_{p\leqslant x}\Delta^{2}(p)

\displaystyle=\frac{\mathcal{C}}{4\pi^{2}}\sum_{p\leqslant x}p^{1/2}+O(x^{23/16+\varepsilon}).

(4.14)

Thus we have completed the proof of the Theorem.

Acknowledgement

The authors would like to appreciate the referee for his/her patience in refereeing this paper. This work is supported by the Beijing Natural Science Foundation(Grant No.1242003), and the National Natural Science Foundation of China (Grant No.11971476).

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	$\displaystyle x^{-\varepsilon}S_{I}$	$\displaystyle\ll\mathop{\sum_{m_{1}\sim M_{1}}\sum_{m_{2}\sim M_{2}}}_{\begin{subarray}{c}m_{1}\neq m_{2}\end{subarray}}\sum_{h\sim H}\left\|\sum_{\ell\sim L}\mathbf{e}(f(\ell))\right\|$
		$\displaystyle\ll\mathop{\sum_{m_{1}\sim M_{1}}\sum_{m_{2}\sim M_{2}}}_{\begin{subarray}{c}m_{1}\neq m_{2}\end{subarray}}\sum_{h\sim H}\left(L(L^{-3/2}H^{1/2}\|\sqrt{m_{1}}-\sqrt{m_{2}}\|)^{1/2}+(L^{-3/2}H^{1/2}\|\sqrt{m_{1}}-\sqrt{m_{2}}\|)^{-1/2}\right)$
		$\displaystyle\ll x^{1/4}H{M_{1}}^{5/4}M_{2}+x^{3/4}\mathop{\sum_{m_{1}\sim M_{1}}\sum_{m_{2}\sim M_{2}}}_{\begin{subarray}{c}m_{1}\neq m_{2}\end{subarray}}\|\sqrt{m_{1}}-\sqrt{m_{2}}\|^{-1/2},$

On a sum of the error term of the Dirichlet divisor function over primes

Abstract.

1. Introduction and Main Result

Theorem.

2. Priliminary

2.1. Notations

2.2. Auxiliary Lemmas

Lemma 2.1.

Proof.

Lemma 2.2.

Proof.

Lemma 2.3.

Proof.

Lemma 2.4.

Proof.

Lemma 2.5.

Proof.

Lemma 2.6.

Proof.

Lemma 2.7.

Proof.

Lemma 2.8.

Proof.

Lemma 2.9.

Proof.

Lemma 2.10.

Proof.

Lemma 2.11.

Proof.

Lemma 2.12.

Proof.

Lemma 2.13.

Proof.

Lemma 2.14.

Proof.

3. The spacing problem

Proposition 3.1.

Proof.

4. Proof of the Theorem

4.1. Estimate of R1R_{1}

Case 1

Case 2

Case 3

Case 4

4.2. Estimate of R2R_{2}

Acknowledgement

References

4.1. Estimate of $R_{1}$

4.2. Estimate of $R_{2}$