16 The generalized Dirichlet divisor problem

For any fixed integer \(k \ge 1\), let

\[ d_k(n) := \sum _{n_1 \cdots n_k = n}1 \]

denote the number of ways a positive integer \(n\) may be written as a product of exactly \(k\) positive integers. The divisor sum

\[ D_k(x) := \sum _{n \leq x} d_k(n) \]

is known to satisfy the asymptotic formula

\[ D_k(x) = x P_{k - 1}(\log x) + \Delta _k(x) \]

where \(P_{k - 1}\) is an explicit polynomial of degree \(k-1\) and \(\Delta _k(x) = o(x)\) is an error term. The (generalized) Dirichlet divisor problem concerns bounding the growth rate of \(\Delta _k(x)\) as \(x\to \infty \).

Definition 16.1 Divisor sum exponents

Let \(k \geq 1\) be a fixed integer. Then, \(\alpha _k\) is the least (fixed) exponent for which

\[ \Delta _k(x) \ll x^{\alpha _k+o(1)} \]

for unbounded \(x {\gt} 0\). Furthermore, \(\beta _k\) is the least (fixed) exponent for which

\[ \left(\frac{1}{x}\int _1^x (\Delta _k(t))^2\text{d}t\right)^{1/2} \ll x^{\beta _k + o(1)} \]

for unbounded \(x {\gt} 0\) (in both definitions, the implied constant may depend on \(k\)).

One can also give a non-asymptotic definition: \(\alpha _k\), \(\beta _k\) are respectively the least exponent such that for all \(\varepsilon {\gt} 0\), there exists \(C = C(\varepsilon , k) {\gt} 0\) for which

\[ |\Delta _k(x)| \leq C x^{\alpha _k + \varepsilon },\qquad (x \ge C) \]

and

\[ \left|\frac{1}{x}\int _1^x(\Delta _k(t))^2\text{d}t\right|^{1/2} \le Cx^{\beta _k + \varepsilon },\qquad (x \ge C). \]

In the case \(k = 1\), the problem is trivial. In particular:

Lemma 16.2 \(d_1\) exponent

One has \(\alpha _1=\beta _1=0\).

Proof ▶

Follows from \(\sum _{n \le x}1 = x + O(1)\).

However, the value of \(\alpha _k\) is not known for \(k \ge 2\). On the other hand, the values of \(\beta _2\) and \(\beta _3\) are known.

Theorem 16.3 Hardy [ 94 ]

One has \(\beta _2 = 1/4\).

Theorem 16.4 Cramér [ 47 ]

One has \(\beta _3 = 1/3\).

Nevertheless, the value of \(\beta _k\) is not known for \(k \ge 4\). Unconditionally, the following lower-bounds are known to hold.

Lemma 16.5 Lower bound on \(\alpha _k\) and \(\beta _k\)

For all \(k \geq 1\), one has

\[ \alpha _k \geq \beta _k \geq \frac{1}{2} - \frac{1}{2k}. \]

Proof ▶

The first inequality follows from inserting the bound \(\Delta _k(x) \ll x^{\alpha _k + o(1)}\) into the definition of \(\beta _k\). The second inequality is due to Titchmarsh [ 275 ] . Note also that the weaker inequality \(\alpha _k \ge 1/2 - 1/(2k)\) was first proved by Hardy [ 93 ] .

It is conjectured that this lower bound on \(\alpha _k\) and \(\beta _k\) is in fact an equality [ 277 , p. 320 ] . Amongst other consequences, this conjecture implies the Lindelöf hypothesis [ 277 , Chapter XII ] .

Heuristic 16.6 Generalised Dirichlet divisor problem conjecture

For all \(k \geq 1\), one has

\[ \alpha _k = \beta _k = \frac{1}{2} - \frac{1}{2k}. \]

The remainder of this chapter focuses on upper bounds on \(\alpha _k\) and \(\beta _k\).

16.1 Known pointwise bounds on divisor sum exponents

Currently the sharpest known upper bound on \(\alpha _2\) is:

Theorem 16.7

[ 186 , Theorem 1.2 ] One has \(\alpha _2 \leq \alpha ^* = 0.314483\ldots \), where \(\alpha ^*\) is the solution to the equation

\[ \frac{8}{25}\alpha - \frac{(\sqrt{2(1+14\alpha )} - 5\sqrt{-1+8\alpha })^2}{200} + \frac{51}{200} = \alpha \]

on the interval \(\alpha \in [0.3, 0.35]\).

Table 16.1 records the historical progression of upper bounds on \(\alpha _2\).

Table 16.1 Historical bounds on \(\alpha _2\)

Reference	Upper bound on \(\alpha _2\)
Dirichlet (1849) [ , Piltz [	\(1/2 = 0.5\)
Voronoi (1903) [ 286 ]	\(1/3 = 0.3333\ldots \)
van der Corput (1922) [ 45 ]	\(33/100 = 0.33\)
van der Corput (1928) [ 46 ]	\(27/82 = 0.3292\ldots \)
Chih (1950) [ 39 ] , Richert (1953) [ 247 ]	\(15/46 = 0.3260\ldots \)
Kolesnik (1969) [ 169 ]	\(12/37 = 0.3243\ldots \)
Kolesnik (1973) [ 170 ]	\(346/1067 = 0.3242\ldots \)
Kolesnik (1982) [ 172 ]	\(35/108 = 0.3240\ldots \)
Kolesnik (1985) [ 173 , p. 118 ]	\(139/429 = 0.3240\ldots \)
Iwaniec–Mozzochi (1988) [ 150 ]	\(7/22 = 0.3181\ldots \)
Huxley (1993) [ 128 ]	\(23/73 = 0.3150\ldots \)
Huxley (2003) [ 130 ]	\(131/416 = 0.3149\ldots \)
Li–Yang (2023) [ 186 ]	\(0.314483\ldots \)

Currently, the sharpest known bound on \(\alpha _3\) is:

Theorem 16.8

[ 171 ] One has \(\alpha _3 \leq 43/96\).

Table 16.2 records the historical progression of upper bounds on \(\alpha _3\).

Table 16.2 Historical bounds on \(\alpha _3\)

Reference	Upper bound on \(\alpha _3\)
Walfisz (1926) [ 288 ]	\(43/87 = 0.4942\ldots \)
Atkinson (1941) [ 1 ]	\(37/75 = 0.4933\ldots \)
Rankin (1955) [ 245 ]	\(0.4931466\ldots \)
Yue (1958) [ 313 ]	\(14/29 = 0.4827\ldots \)
Yin (1959) [ 308 ]	\(25/52 = 0.4807\ldots \)
Yin (1959) [ 309 ]	\(10/21 = 0.4761\ldots \)
Yue–Wu (1962) [ 314 ]	\(8/17 = 0.4705\ldots \)
Chen (1965) [ 33 ]	\(5/11 = 0.4545\ldots \)
Yin (1964) [ 310 ]	\(34/75 = 0.4533\ldots \)
Yin–Li (1981) [ 311 ] , Zheng (1988) [ 317 ]	\(127/282 = 0.4503\ldots \)
Kolesnik (1981) [ 171 ]	\(43/96 = 0.4479\ldots \)

For larger \(k\), estimates typically make use of the following relationship with zeta-moments.

Lemma 16.9

Let \(k \geq 2\) be an integer. If \(M(\sigma ,k) = 1\) then \(\alpha _k \leq \sigma \).

Proof ▶

See [ 144 , § 13.3 ] .

For completeness we record the historical progression in bounds for \(\alpha _k\).

Lemma 16.10 Piltz bound

For \(k \ge 2\), one has

\[ \alpha _k \le 1 - \frac{1}{k}. \]

Lemma 16.11 Voronoi, Landau bound

For \(k \ge 2\), one has

\[ \alpha _k \leq 1 - \frac{2}{k + 1}. \]

Proof ▶

See Voronoi [ 286 ] for \(k = 2\) and Landau [ 176 ] for \(k \ge 3\).

Lemma 16.12 Hardy–Littlewood bound for \(k \ge 4\)

For \(k \ge 4\), one has

\[ \alpha _k \leq 1 - \frac{3}{k + 2}. \]

Proof ▶

See [ 96 ] . The original proof relied on the assumption that \(\mu (1/2) \le 1/6\) which was published later.

Lemma 16.13 Tong bound for \(4 \le k \le 11\)

One has

\begin{alignat*}{4} \alpha _4 & \le 1/2,\qquad & & \alpha _5 \le 4/7,\qquad & & \alpha _6 \le 5/8,\qquad & & \alpha _7 \le 71/107\\ \alpha _8 & \le 41/59,\qquad & & \alpha _9 \le 31/43,\qquad & & \alpha _{10} \le 26/35,\qquad & & \alpha _{11}\le 19/25 \end{alignat*}

Proof ▶

See Tong [ 278 ] .

Theorem 16.14

[ 108 ] For \(4 \le k \le 8\), one has

\[ \alpha _k \leq \frac{3k-4}{4k}. \]

Theorem 16.15 Ivić–Ouellet bound for large \(k\)

[ 145 ] One has

\begin{align*} \alpha _{10} \le 27/40,\qquad \alpha _{11} & \le 0.6957,\qquad \alpha _{12} \le 0.7130,\qquad \alpha _{13} \le 0.7306,\\ \alpha _{14} \le 0.7461,\qquad \alpha _{15} & \le 0.75851,\qquad \alpha _{16} \le 0.7691,\qquad \alpha _{17} \le 0.7785,\\ \alpha _{18} \le 0.7868,\qquad \alpha _{19} & \le 0.7942,\qquad \alpha _{20} \le 0.8009. \end{align*}

Theorem 16.16

[ 144 , Theorem 13.12 ] One can bound \(\alpha _k\) by

\begin{align*} (3k-4)/4k & \hbox{ for } 4 \leq k \leq 8 \\ 35/54 & \hbox{ for } k = 9 \\ 41/60 & \hbox{ for } k = 10 \\ 7/10 & \hbox{ for } k = 11 \\ (k-2)/(k+2) & \hbox{ for } 12 \leq k \leq 25 \\ (k-1)/(k+4) & \hbox{ for } 26 \leq k \leq 50 \\ (31k-98)/32k & \hbox{ for } 51 \leq k \leq 57 \\ (7k-34)/7k & \hbox{ for } k \geq 58. \end{align*}

Lemma 16.17 Heath-Brown bound for large \(k\)

For any \(k \ge 2\), one has

\[ \alpha _k \le 1 - 0.849k^{-2/3}. \]

Proof ▶

See Heath-Brown [ 113 ] .

Theorem 16.18 [ 15 ]

For integer \(k \ge 30\), one has

\[ \alpha _k \leq 1 - 1.421(k - 1.18)^{-2/3}. \]

Moreover, \(\alpha _k \leq 1 - 1.889k^{-2/3}\) for sufficiently large \(k\).

Theorem 16.19 Trudgian–Yang bound for large \(k\)

[ [ 279 ] , Theorem 2.9]One has

\begin{align*} \alpha _{9} \le 0.64720,\qquad \alpha _{10} \le 0.67173,\qquad \alpha _{11} & \le 0.69156,\qquad \alpha _{12} \le 0.70818,\\ \alpha _{13} \le 0.72350, \qquad \alpha _{14} \le 0.73696,\qquad \alpha _{15} & \le 0.74886,\qquad \alpha _{16} \le 0.75952,\\ \alpha _{17} \le 0.76920, \qquad \alpha _{18} \le 0.77792,\qquad \alpha _{19} & \le 0.78581,\qquad \alpha _{20} \le 0.79297,\\ \alpha _{21} \le 0.79951. \end{align*}

Theorem 16.20 Li bound for large \(k\)

[ [ 184 ] , Theorem 2]One has

\begin{align*} \alpha _{9} \le 0.638889,\qquad \alpha _{10} \le 0.663329,\qquad \alpha _{11} & \le 0.684349,\qquad \alpha _{12} \le 0.701768,\\ \alpha _{13} \le 0.717523, \qquad \alpha _{14} \le 0.731898,\qquad \alpha _{15} & \le 0.744898,\qquad \alpha _{16} \le 0.75638,\\ \alpha _{17} \le 0.766588, \qquad \alpha _{18} \le 0.775721,\qquad \alpha _{19} & \le 0.783939,\qquad \alpha _{20} \le 0.791374. \end{align*}