On mean square of the error term of a multivariable divisor function

1.   Introduction

Let $\tau (n)$ be the Dirichlet divisor function. It is known that for a real number $x\geqslant2$,

where $\gamma$ is Euler's constant. It was first proved by Dirichlet that

Let $\theta$ denotes the smallest number such that

holds for any $\varepsilon > 0$. Many authors worked on making $\theta$ smaller, such as Voronoi ^[1], Corput ^[2], Kolesnik ^[3], Huxley ^[4], etc. Until now the best result, namely $\theta = 131/416$ is due to Huxley ^[4].

It is conjectured that $\theta = 1/4$, which is supported by the classical mean square result: Suppose $T$ is a large real number, then for any $\varepsilon > 0$, one has

where

This result was given by Cramér ^[5] in 1922. In 1956, Tong ^[6] showed that the error term in (1.3) can be reduced to $T\log^5T$. In 1988, Preissmann ^[7] reduced the error term to $T\log^4T$. In 2009, Lau and Tsang ^[8] proved that

Let $k\geqslant2$ be a fixed integer. Tóth and Zhai ^[9] considered the average of divisor function of $k$ variables. They obtain the asymptotic formula

holds for any $\varepsilon > 0$, where $\theta$ is the exponent in (1.2) and $P_k(t)$ is a polynomial in $t$ of degree $k$. We denote $\Delta_{k}(x)$ by

We have the following theorem:

Theorem 1.1. Suppose $T\geqslant2$ is a large real number, then we have

where $L_{2k-2}(u)$ is a polynomial in $u$ of degree $2k-2$, and the implied constant about "$O$" depends on $k$ and $\varepsilon$.

2.   Some lemmas

We present some lemmas.

Lemma 2.1. Suppose $x\geqslant2$ is large, then for any $1\ll N\ll x$, we have

Proof. See [10,Chapter 3.2]. □

Lemma 2.2. Suppose $T\geqslant2$, $T^{\varepsilon}\ll y\ll T$, and let

then we have

Proof. See, for example, the following references: Lau and Tsang ^[8], Tsang ^[11], Zhai ^[12]. □

Lemma 2.3. Suppose $G_0, m_0$ are fixed real positive numbers, let G(x) be a monotonic function defined on $[a, b]$ such that

and m(x) be a differentiable real function such that

on $[a, b]$, $F(\cdot) = \cos(\cdot)$ or $\sin(\cdot)$ or $e(\cdot)$, then

Proof. See [10,Lemma 2.1]. □

Lemma 2.4. Let $k\geqslant2$ be a fixed integer and let $s_1, \cdots, s_k$ be complex numbers. Then for $\Re{s_j} > 1$, where $j = 1, 2, \cdots, k$, we have

where

This series is absolutely convergent provided that $\Re s_j > 0$ and $\Re(s_j+s_l) > 1 (1\leqslant j, l\leqslant k)$, and $f(n_1, \cdots, n_k)$ is multiplicative and symmetric in all variables.

Moreover, for $r_1, \cdots, r_k\in\{1, 2\}$,

is absolutely convergent provided that $\Re s_j > 0$ and $\Re(s_j+s_l) > 1 (1\leqslant j, l\leqslant k)$.

Proof. See Tóth and Zhai [9,Proposition 2.1], and the convergence of (2.1) is a direct corollary. □

Lemma 2.5. Suppose $x, y$ are large real numbers, $k\geqslant2$ is a fixed integer, $s, w$ are given real numbers such that $0 < s < 1/2 < w < 1$, $f$ is defined in Lemma 2.4. Let $\mathbf{M}_1$ and $\mathbf{M}_2$ denote the vectors $(m_1, \cdots, m_{k})$ and $(m_{k+1}, \cdots, m_{2k})$, $D_1$ and $D_2$ denote $\left(\prod_{j = 1}^{k-1}m_j\right)$ and $\left(\prod_{j = k+1}^{2k-1}m_j\right)$, respectively. Let

where $g(\mathbf{M}_1, \mathbf{M}_2)$ is any function that satisfies

then we have:

(i) $T_{g, k}(s, w)$ is absolutely convergent.

(ii) We have

Proof. (i) Since ${m_k}/{m_{2k}} = {n_1}/{n_2}$, we can find positive integers $t_1, t_2, g_1, g_2$ such that

where $(t_1, t_2) = 1$ and $t_1g_1 = m_k$, $t_2g_1 = m_{2k}$, $t_1g_2 = n_1$, $t_2g_2 = n_2$. Denote by ${\mathbf{M}_1}^{'}$ and ${\mathbf{M}_2}^{'}$ the vectors $(m_1, \cdots, m_{k-1})$ and $(m_{k+1}, \cdots, m_{2k-1})$, respectively, then we have

where we use the conclusion: For a given real number $c$, we have

moreover, for a large real number $U$, using partial summation and the conclusion

we have

Since for any real $\delta > 0$,

then

Let $t_1g_1 = m_k$, $t_2{g_1}^{'} = {m_{2k}}^{'}$, by Lemma 2.4, we have

Thus, we conclude that (i) holds.

(ii) Since $m_1, \cdots, m_{k-1}, m_{k+1}, \cdots, m_{2k-1}$ are symmetric, we have

where

Similar to (i), let $(t_1, t_2) = 1$ and $t_1g_1 = m_k$, $t_2g_1 = m_{2k}$, $t_1g_2 = n_1$, $t_2g_2 = n_2$, and denote by ${\mathbf{M}_1}^{'}$ and ${\mathbf{M}_2}^{'}$ the vectors $(m_1, \cdots, m_{k-1})$ and $(m_{k+1}, \cdots, m_{2k-1})$ respectively, then

where we use (2.2).

Let

by (2.4), we have

Let $t_1g_1 = m_k$, $t_2{g_1}^{'} = {m_{2k}}^{'}$, we have

the convergence of the two series in the last step can be obtained by Lemma 2.4, and we use the arbitrariness of $\varepsilon$.

For $T_2$, if $m_k > x$, we replace the condition "$m_1 > x$" by "$t_1g_1 > x$" in (2.6), thus

Using (2.4) we have

Let $t_1g_1 = m_k$, $t_2{g_1}^{'} = {m_{2k}}^{'}$, we have

the convergence of the two series in the last step can be obtained by Lemma {2.4}, and we use the arbitrariness of $\varepsilon$. If $m_{2k} > x$, exchange $t_1$ and $t_2$, we can also obtain

For $T_3$, if $n_1 > y$, similar to (i), we obtain $(t_1, t_2) = 1$ and $t_1g_1 = m_k$, $t_2g_1 = m_{2k}$, $t_1g_2 = n_1$, $t_2g_2 = n_2$, then

where we use (2.3) by taking $U = y/t_1$.

We denote the latter series by $\Sigma$, using (2.4) we have

Let $t_1g_1 = m_k$, $t_2{g_1}^{'} = {m_{2k}}^{'}$, we obtain

by Lemma 2.4, thus

If $n_2 > y$, exchange $t_1$ and $t_2$, similarly we obtain

Above all, we obtain

then (ii) holds from (2.5). □

3.   Expression of ${\Delta_k}(x)$

We shall give a more explicit expression of ${\Delta_k}(x)$. According to Lemma 2.4,

holds for any $n_1, \cdots, n_k\in \mathbb{N}$, where $f$ is multiplicative and symmetric in all variables.

Therefore, we deduce by (1.1) that

where

Here we have

by the proof of Theorem 3.4 in Tóth and Zhai ^[9]. From (1.2) we have

for any $1\leqslant j\leqslant k$, then

where the convergence of latter series can be obtained by Lemma 2.4. Then we conclude that

follows by (1.4), (3.1), (3.3), and (3.4), where

4.   Mean square of ${\Delta_k}^{*}(x)$

Suppose $T\geqslant2$, we shall first estimate $\int_T^{2T}({\Delta_k}^{*}(x))^2dx$. Since $m_1, \cdots, m_{k-1}$ are symmetric, we can divide ${\Delta_k}^{*}(x)$ into three parts,

where the implied constant about "$O$" depends on $k$ and $\varepsilon$,

where $y$ is a parameter that satisfies $T^{\varepsilon}\ll y\ll T$. So we obtain

4.1.   Mean square of $M_1(x, y)$

First, we deal with $\int_{T}^{2T}{M_1}^2dx$. By Lemma 2.1 we obtain

where

and $\delta_1(\cdot, y)$ and $\delta_2(\cdot, y)$ are defined in Lemma 2.2, thus we have

Let $\mathbf{M_1}, \mathbf{M_2}$ be defined in Lemma 2.5, using

we have

where

So, it turns to evaluate

We need to give more explicit expressions for $(\prod_{j = 1}^{k-1}M(\frac{x}{m_j}))$ and $(\prod_{j = k+1}^{2k-1}M(\frac{x}{m_j}))$. By the proof of Theorem 3.3 in Tóth and Zhai ^[9] in the case $f_j(n) = \tau(n), \:a_j = 1, \:\delta_j = 1\:(1\leqslant j\leqslant k-1)$ we obtain

where

and $c(j_1, \cdots, j_{k-1})$ are constants $(j_1, \cdots, j_{k-1} = 0, 1)$. Similarly, when $k+1\leqslant j\leqslant 2k-1$, we have

where

and $c(j_{k+1}, \cdots, j_{2k-1})$ are constants $(j_{k+1}, \cdots, j_{2k-1} = 0, 1)$.

4.1.1.   Evaluation of $\int_0$

We denote $C_{l_1}(\log m_1, \cdots, \log m_{k-1})C_{l_2}(\log m_{k+1}, \cdots, \log m_{2k-1})$ by $C_{l_1, l_2}(\mathbf{M}_1, \mathbf{M}_2)$, using (4.5) and (4.6) we have

where $D_1, D_2$ are defined in Lemma 2.5. Using (2.4) we obtain

Choosing

$s = 1/4$, $w = 3/4$ in Lemma 2.5, we obtain that $S_0(x, y)$ can be written as

Since $g(\mathbf{M}_1, \mathbf{M}_2)$ is related to $l_1, l_2$, we denote $T_{g, k}\left(\frac{1}{4}, \frac{3}{4}\right)$ in (4.8) by $D_{k, l_1, l_2}$, we conclude that

where

Then it follows from (4.9) that

4.1.2.   Estimates of $\int_1$ and $\int_2$

Let $\mathbf{M_1}, \mathbf{M_2}, D_1, D_2$ be defined in Lemma 2.5, then by (4.5) and (4.6) we have

Let

in Lemma 2.3, change the order of integration and summation, then we have

then we use

to obtain

where by partial summation we obtain

Since $\varepsilon > 0$ is arbitrary, using Lemma 2.4 we conclude that

Then we turn to estimate $\int_1$, similar to $\int_2$, we have

Let

in Lemma 2.3, change the order of integration and summation, then we have

Since

we have

where

Then we have

where we use partial summation to obtain

And for $R_1$, by Lagrange's mean value theorem we have

for any

thus let

in $R_1$, we obtain

For some real numbers $N_1$, $N_2$ satisfying $1\leqslant N_1, N_2\leqslant y$, one has

we denote the latter sum by $T(m_k, m_{2k})$. Let

we have

so we can find a constant $c_0$ such that

where $t$ is an integer such that $0 < t < 2y^2$, thus

similarly, we have

thus

By (4.14) we have

Finally by (4.12), (4.13) and (4.15) we obtain

thus

since $\varepsilon > 0$ is arbitrary, $T^\varepsilon\ll y\ll T$, and the convergence of the latter series can be obtained by Lemma 2.4. Above all, by (4.4), (4.10), (4.11), and (4.16) we obtain

4.1.3.   Other terms in (4.3)

We are going to estimate $\int_T^{2T}M_{12}(x, y)dx$,

By Lemma 2.2 we have

similarly, we have

Thus,

since $y\ll T$, and the convergence of the latter series is given by Lemma 2.4.

Then by (4.17), (4.19), and Cauchy-Schwarz's inequality, we are able to obtain

So from (4.3) and (4.17)–(4.20), we obtain

4.2.   Error terms in (4.2)

Let $\mathbf{M_1}, \mathbf{M_2}, D_1, D_2$ be defined in Lemma 2.5, then for $\int_T^{2T}{M_3}^2dx$ we deduce that

change order of integration and summation we obtain

Using Cauchy-Schwarz's inequality and (1.3), we deduce that

thus, by Lemma 2.4, we have

For $\int_T^{2T}{M_2}^2dx$, let ${D_1}^{'} = D_1/m_1$, ${D_2}^{'} = D_2/m_{k+1}$, similar to $\int_T^{2T}{M_3}^2dx$, we obtain by Lemma 2.4 that

By (4.21)–(4.23) and Cauchy-Schwarz's inequality, we are able to estimate the following terms in (4.2)

Above all, taking $y = T^{\frac{2}{5}}$, then

follows by (4.2) and (4.21)–(4.24).

5.   Proof of the theorem

Using (3.5) and the Cauchy-Schwarz's inequality we obtain

Then replacing $T$ by $T/2$, $T/2^2$, and so on, and adding up all the results, we obtain

where we use integration by part several times to obtain $L_{2k-2}(u)$ is a polynomial in $u$ of degree $2k-2$ denoted by

To sum up, this finishes the proof of the Theorem.

6.   Conclusions

In this paper, we give an asymptotic formula of the mean square of $\Delta_{k}(x)$, which can be viewed as an analogue of (1.3). We use the convergence of the multivariable Dirichlet series, and it can be used to show the properties of other multivariable arithmetic functions. In 2023, Tóth^[13], Heyman and Tóth^[14] gave some useful applications of the Dirichlet series.

Acknowledgments

The author would like to appreciate the referee for his/her patience in refereeing this paper. This work is supported by Natural Science Foundation of Beijing Municipal (Grant No.1242003), and the National Natural Science Foundation of China (Grant Nos.12471009 and 12301006).

Conflict of interest

No potential conflicts of interest were reported by the author.