On the Generalized Dirichlet Divisor Problem

1. Introduction

Let $k⩾2$ denotes an integer and $d_k\left(n\right)$ is the divisor function that represents the number of ways n may be written as a product of exactly k factors. The generalized Dirichlet divisor problem consists of the estimation of the function

$${\Delta }_k\left(x\right)=\sum _{n⩽x}{d}_k\left(n\right)-x{P}_{k-1}(logx),$$

(1)

where $P_{k-1}$ is an explicit polynomial of degree $k-1$. Clearly we have ${\Delta }_k\left(x\right)=o\left(x\right)$. We then define ${\alpha }_k$ as the least exponent for which

$${\Delta }_k\left(x\right)\ll x^{{\alpha }_k+\epsilon }.$$

(2)

In 1916, Hardy [1] first proved a lower bound that ${\alpha }_k⩾{\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{2k}}$ for all $k⩾2$. The generalized Dirichlet divisor problem conjecture states that ${\alpha }_k={\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{2k}}$ holds for all $k⩾2$, and this conjecture implies the Lindelöf hypothesis. Now, the best upper bounds for ${\alpha }_k$ ($k⩽8$) are

$${\alpha }_2⩽0.3144831759741,{\alpha }_3⩽{\displaystyle \frac{43}{96}},{\alpha }_k⩽{\displaystyle \frac{3k-4}{4k}}\mathrm{for}4⩽k⩽8$$

by Li and Yang [2], Kolesnik [3] and Heath–Brown [4] (and Ivić [5]) respectively. Ivić also gave upper bounds with $k⩾9$ in his book. For results with large k, one can see works of Heath–Brown [6] and Bellotti and Yang [7]. We also refer the readers to the blueprint of the new project ANTEDB organized by Tao, Trudgian and Yang [8].

In 1989, Ivić and Ouellet [9] refined the technique used in and gave better bounds for ${\alpha }_k$ with $k⩾9$. In [5], Ivić connected this problem with the function $m\left(\sigma \right)$ defined as follows: For any fixed ${\displaystyle \frac{1}{2}}<\sigma <1$ we define $m\left(\sigma \right)$ as the supremum of all numbers $m⩾4$ such that

$${\int }_1^T{\left|\zeta (\sigma +it)\right|}^{m}dt\ll T^{1+\epsilon }.$$

(3)

In order to obtain good bounds for ${\alpha }_k$, one need to get lower bounds for $m\left(\sigma \right)$. Ivić and Ouellet [9] used a large value theorem and growth exponents for Riemann zeta–function to bound $m\left(\sigma \right)$. Specially, for $10⩽k⩽20$ they got

$$\begin{array}{cc}\hfill {\alpha }_{10}⩽0.675,{\alpha }_{11}⩽0.6957,{\alpha }_{12}& ⩽0.7130,{\alpha }_{13}⩽0.7306,\hfill \\ \hfill {\alpha }_{14}⩽0.7461,{\alpha }_{15}⩽0.75851,{\alpha }_{16}& ⩽0.7691,{\alpha }_{17}⩽0.7785,\hfill \\ \hfill {\alpha }_{18}⩽0.7868,{\alpha }_{19}⩽0.7942,{\alpha }_{20}& ⩽0.8009.\hfill \end{array}$$

In 2024, Trudgian and Yang [10] mentioned a series of new bounds for ${\alpha }_k$. They combined the method of Ivić and Ouellet [9] with their new growth exponents for Riemann zeta–function to obtain those bounds.

Theorem 1.1 

([10], Theorem 2.9). We have

$$\begin{array}{cc}\hfill {\alpha }_9⩽0.64720,{\alpha }_{10}⩽0.67173,{\alpha }_{11}& ⩽0.69156,{\alpha }_{12}⩽0.70818,\hfill \\ \hfill {\alpha }_{13}⩽0.72350,{\alpha }_{14}⩽0.73696,{\alpha }_{15}& ⩽0.74886,{\alpha }_{16}⩽0.75952,\hfill \\ \hfill {\alpha }_{17}⩽0.76920,{\alpha }_{18}⩽0.77792,{\alpha }_{19}& ⩽0.78581,{\alpha }_{20}⩽0.79297,{\alpha }_{21}⩽0.79951.\hfill \end{array}$$

In this paper, we use the essentially same methods to give a very minor improvement on their results.

Theorem 1.2. 

We have

$$\begin{array}{cc}\hfill {\alpha }_9⩽0.638889,{\alpha }_{10}⩽0.663329,{\alpha }_{11}& ⩽0.684349,{\alpha }_{12}⩽0.701768,\hfill \\ \hfill {\alpha }_{13}⩽0.717523,{\alpha }_{14}⩽0.731898,{\alpha }_{15}& ⩽0.744898,{\alpha }_{16}⩽0.756380,\hfill \\ \hfill {\alpha }_{17}⩽0.766588,{\alpha }_{18}⩽0.775721,{\alpha }_{19}& ⩽0.783939,{\alpha }_{20}⩽0.791374.\hfill \end{array}$$

2. Growth Exponents for Riemann Zeta–Function

In this section we list the new growth exponents for Riemann zeta–function proved by Trudgian and Yang [10], which is the most powerful and important input in the proof of Theorem 1.2. In the proof of Theorem 1.1 a weaker version of this was used.

Lemma 2.1 

([8], Table 6.2). We have

$$\begin{array}{c}\hfill \mu \left(\sigma \right)⩽\left\{\begin{array}{cc}{\displaystyle \frac{31}{36}}-{\displaystyle \frac{3}{7}}\sigma ,\hfill & {\displaystyle \frac{1}{2}}⩽\sigma ⩽{\displaystyle \frac{88225}{153852}},\hfill \\ {\displaystyle \frac{220633}{620612}}-{\displaystyle \frac{62831}{155153}}\sigma ,\hfill & {\displaystyle \frac{88225}{153852}}⩽\sigma ⩽{\displaystyle \frac{521}{796}},\hfill \\ {\displaystyle \frac{1333}{3825}}-{\displaystyle \frac{1508}{3825}}\sigma ,\hfill & {\displaystyle \frac{521}{796}}⩽\sigma ⩽{\displaystyle \frac{53141}{76066}},\hfill \\ {\displaystyle \frac{405}{1202}}-{\displaystyle \frac{227}{601}}\sigma ,\hfill & {\displaystyle \frac{53141}{76066}}⩽\sigma ⩽{\displaystyle \frac{454}{641}},\hfill \\ {\displaystyle \frac{779}{2590}}-{\displaystyle \frac{423}{1295}}\sigma ,\hfill & {\displaystyle \frac{454}{641}}⩽\sigma ⩽{\displaystyle \frac{1744}{2411}},\hfill \\ {\displaystyle \frac{179}{622}}-{\displaystyle \frac{96}{311}}\sigma ,\hfill & {\displaystyle \frac{1744}{2411}}⩽\sigma ⩽{\displaystyle \frac{951057}{1298878}},\hfill \\ {\displaystyle \frac{157319}{560830}}-{\displaystyle \frac{251324}{841245}}\sigma ,\hfill & {\displaystyle \frac{951057}{1298878}}⩽\sigma ⩽{\displaystyle \frac{1389}{1736}},\hfill \\ {\displaystyle \frac{2841}{10316}}-{\displaystyle \frac{754}{2579}}\sigma ,\hfill & {\displaystyle \frac{1389}{1736}}⩽\sigma ⩽{\displaystyle \frac{587779}{702192}},\hfill \\ {\displaystyle \frac{1691}{6554}}-{\displaystyle \frac{890}{3277}}\sigma ,\hfill & {\displaystyle \frac{587779}{702192}}⩽\sigma ⩽{\displaystyle \frac{7441}{8695}},\hfill \\ {\displaystyle \frac{29}{130}}-{\displaystyle \frac{3}{13}}\sigma ,\hfill & {\displaystyle \frac{7441}{8695}}⩽\sigma ⩽{\displaystyle \frac{277}{300}},\hfill \\ {\displaystyle \frac{3}{23}}-{\displaystyle \frac{3}{23}}\sigma ,\hfill & {\displaystyle \frac{277}{300}}⩽\sigma <1.\hfill \end{array}\right.\end{array}$$

In order to use the large value theorem in the next section, we also need the following two results, which give the upper bounds for $\mu \left(\sigma \right)$ when $\sigma <{\displaystyle \frac{1}{2}}$:

Lemma 2.2 

([8], Lemma 6.4). We have

$$\mu \left(\sigma \right)=\mu (1-\sigma )+{\displaystyle \frac{1}{2}}-\sigma$$

for all $0<\sigma ⩽{\displaystyle \frac{1}{2}}$.

Lemma 2.3 

([8], Lemma 6.5). We have

$$\mu \left(\sigma \right)={\displaystyle \frac{1}{2}}-\sigma$$

for all $\sigma ⩽0$.

3. A Large Value Theorem

Now we provide a new large value theorem, which is a refined version of Ivić’s large value theorem [[5], Lemma 8.2]. Note that Ivić’s version was used by Ivić and Ouellet [9].

Lemma 3.1. 

Let $t_1,\dots ,t_R$ be real numbers such that $T⩽t_r⩽2T$ for $r=1,\dots ,R$ and $|t_r-t_s{|⩾(logT)}^4$ for $1⩽r\ne s⩽R$. If

$$T^{\epsilon }<V⩽\left|\sum _{m\sim M}{a}_m{m}^{-\sigma -i{t}_r}\right|$$

where $a_m\ll M^{\epsilon }$ for $m\sim M$, $1\ll M\ll T^C$, then

$$R\ll T^{\epsilon }\left(M^{2-2\sigma }{V}^{-2}+T{V}^{-f\left(\sigma \right)}\right),$$

where

$$\begin{array}{c}\hfill f\left(\sigma \right)=\left\{\begin{array}{cc}{\displaystyle \frac{2}{3-4\sigma }},\hfill & {\displaystyle \frac{1}{2}}<\sigma ⩽{\displaystyle \frac{2}{3}},\hfill \\ {\displaystyle \frac{58}{63-80\sigma }},\hfill & {\displaystyle \frac{2}{3}}<\sigma ⩽{\displaystyle \frac{583}{860}},\hfill \\ {\displaystyle \frac{47}{41-50\sigma }},\hfill & {\displaystyle \frac{583}{860}}<\sigma ⩽{\displaystyle \frac{16581}{24022}},\hfill \\ {\displaystyle \frac{4968}{3981-4774\sigma }},\hfill & {\displaystyle \frac{16581}{24022}}<\sigma ⩽{\displaystyle \frac{1333047}{1920826}},\hfill \\ {\displaystyle \frac{15998}{12283-14600\sigma }},\hfill & {\displaystyle \frac{1333047}{1920826}}<\sigma ⩽{\displaystyle \frac{3269}{4658}},\hfill \\ {\displaystyle \frac{656601}{497599-589921\sigma }},\hfill & {\displaystyle \frac{3269}{4658}}<\sigma ⩽{\displaystyle \frac{2410373}{3361430}},\hfill \\ {\displaystyle \frac{245}{182-215\sigma }},\hfill & {\displaystyle \frac{2410373}{3361430}}<\sigma ⩽{\displaystyle \frac{2233}{3105}},\hfill \\ {\displaystyle \frac{1037}{743-872\sigma }},\hfill & {\displaystyle \frac{2233}{3105}}<\sigma ⩽{\displaystyle \frac{592}{819}},\hfill \\ {\displaystyle \frac{503}{325-374\sigma }},\hfill & {\displaystyle \frac{592}{819}}<\sigma ⩽{\displaystyle \frac{140323}{193464}},\hfill \\ {\displaystyle \frac{12982}{8109-9268\sigma }},\hfill & {\displaystyle \frac{140323}{193464}}<\sigma ⩽{\displaystyle \frac{1461}{1982}},\hfill \\ {\displaystyle \frac{1061878}{648903-738576\sigma }},\hfill & {\displaystyle \frac{1461}{1982}}<\sigma ⩽{\displaystyle \frac{1960121}{2577906}},\hfill \\ {\displaystyle \frac{146}{85-96\sigma }},\hfill & {\displaystyle \frac{1960121}{2577906}}<\sigma ⩽{\displaystyle \frac{76}{97}},\hfill \\ {\displaystyle \frac{158}{67-72\sigma }},\hfill & {\displaystyle \frac{76}{97}}<\sigma ⩽{\displaystyle \frac{1960121}{2419300}}.\hfill \end{array}\right.\end{array}$$

Proof. 

We follow the arguments in [5], Lemma 8.2]. Let $c\left(\sigma \right)$ be an upper bound for $\mu \left(\sigma \right)$. By Lemmas 2.1–2.3, we know that $c\left(\sigma \right)$ can be written in the form $A-B\sigma$ with potisive $A,B$ when $\sigma ⩽0.8$. We choose $0.8$ as the end point because it is enough for our proof of Theorem 1.2. Let $\theta =\theta \left(\sigma \right)$ be implicitly defined by

$$2c\left(\theta \right)+1+\theta -2(1+c(\theta \left)\right)\sigma =0.$$

(4)

Suppose that for some $\sigma$, the value of $\theta$ lies in an interval $[{\sigma }_1,{\sigma }_2]$ with fixed $A,B$. Then by (4) we have

$$\theta ={\displaystyle \frac{2(1+A)\sigma -(2A+1)}{2B\sigma +(1-2B)}}.$$

(5)

Furthermore, let

$$f\left(\sigma \right)={\displaystyle \frac{2(1+c(\theta \left)\right)}{c\left(\theta \right)}}.$$

(6)

The values of $f\left(\sigma \right)$ when ${\displaystyle \frac{1}{2}}⩽\sigma ⩽0.8$ are listed above. Now by similar arguments as in the proof of [[5], Lemma 8.2], Lemma 3.1 is proved. □

4. Proof of Theorem 1.2

We shall use the method of Ivić and Ouellet [9] to prove Theorem 1.2. It was shown in [5], Chapter 8] that to obtain bounds for $m\left(\sigma \right)$ it suffices to obtain bounds of the form

$$R\ll T^{1+\epsilon }{V}^{-m\left(\sigma \right)},$$

(7)

where R is the number of points $t_r(1⩽r⩽R)$ such that $|t_r|⩽T$, $|t_r-t_s{|⩾(logT)}^4$ for $1⩽r\ne s⩽R$ and $\left|\zeta \right(\sigma +i{t}_r\left)\right|⩾V>0$ for any given V. Moreover, by [5], (8.97)] we know that

$$R\ll T^{\epsilon }\left(T{V}^{-2f\left(\sigma \right)}+T^{{\displaystyle \frac{4-4\sigma }{1+2\sigma }}}{V}^{{\displaystyle \frac{-12}{1+2\sigma }}}+T^{{\displaystyle \frac{4(1-\sigma )(\kappa +\lambda )}{\left(\right(2-4\lambda )\sigma -1+2\kappa -2\lambda )}}}{V}^{{\displaystyle \frac{-4(1+2\kappa +2\lambda )}{\left(\right(2-4\lambda )\sigma -1+2\kappa -2\lambda )}}}\right),$$

(8)

where $(\kappa ,\lambda )$ is an exponent pair. We shall use $(\kappa ,\lambda )=\left({\displaystyle \frac{3}{40}},{\displaystyle \frac{31}{40}}\right)$ in the rest of our paper for the sake of convenience.

Note that $c\left(\sigma \right)$ is an upper bound for $\mu \left(\sigma \right)$ given by Lemmas 2.1–2.3. By (8) and the definitions of $f\left(\sigma \right)$ and $c\left(\sigma \right)$, we can easily calculate the corresponding $m\left(\sigma \right)$ for $\sigma$ between ${\displaystyle \frac{1}{2}}$ and $0.8$. Numerical calculation gives that

$$\begin{array}{cc}\hfill m\left(0.638889\right)>9,m\left(0.663329\right)>10,m\left(0.684349\right)& >11,m\left(0.701768\right)>12,\hfill \\ \hfill m\left(0.717523\right)>13,m\left(0.731898\right)>14,m\left(0.744898\right)& >15,m\left(0.756380\right)>16,\hfill \\ \hfill m\left(0.766588\right)>17,m\left(0.775721\right)>18,m\left(0.783939\right)& >19,m\left(0.791374\right)>20,\hfill \end{array}$$

and Theorem 1.2 is now proved.