Erdős–Hooley Delta-function: Overview

Updated 8 December 2025

The Erdős–Hooley Delta-function is defined as the maximum count of divisors in a sliding exponential window for an integer.
It plays a crucial role in analytic number theory by measuring local divisor clumping and connecting classical divisor functions with refined local measures.
Recent advances have sharpened the mean value and moment bounds using methods like Fourier analysis, combinatorial techniques, and probabilistic models.

The Erdős–Hooley Delta-function $\Delta(n)$ is a central object in the analytic paper of local divisor concentrations of integers. For $n \geq 1$ and real $u$ , define the local divisor count $\Delta(n, u) = |\{d : d \mid n,\ e^u < d \leqslant e^{u+1}\}|$ , measuring the number of divisors of $n$ in the multiplicative window $(e^u, e^{u+1}]$ . The Delta-function is the global maximum: $\Delta(n) = \max_{u \in \mathbb{R}} \Delta(n, u)$ . This function quantifies the maximal "clumping" of divisors of $n$ on the log-scale and is situated, in terms of extremal behaviour, between the divisor function $\tau(n)$ and finer local measures of divisor spacing. Its moments, mean values, distributional limits, and variants twisted by characters on $\mathbb{Z}$ or ideals in number fields, are major topics in probabilistic and combinatorial number theory, with recent advances yielding nearly optimal bounds and conjectures relating to high-multiplicity subset sums.

1. Historical Context and Early Results

The paper of $\Delta(n)$ originates from P. Erdős in the 1970s, seeking quantitative understanding of how divisors concentrate within short intervals on the exponential scale. Hooley proved in 1979 that the average order of $\Delta(n)$ is "substantially smaller than the ordinary divisor function" $\tau(n)$ : $\sum_{n \leq x} \Delta(n) \ll x (\log x)^{4/\pi - 1}$ , a saving over the classical mean value of $\tau(n) \sim x \log x$ (Koukoulopoulos et al., 2023). Subsequently, Hall and Tenenbaum obtained refined exponential and sub-exponential upper bounds— $\sum_{n \leq x} \Delta(n) \ll x \exp((1+\epsilon)\log_2 x \log_3 x)$ and similar estimates—together with early lower bounds of the form $\sum_{n\leq x}\Delta(n)\gg x\log\log x$ (Ford et al., 2023, Bretèche et al., 2022).

Maier and Tenenbaum showed that for almost all $n$ , the normal order satisfies $\Delta(n)\leq (\log_2 n)^{y_2}$ for any $y_2 > \log 2 \approx 0.6931$ , and Ford, Green, and Koukoulopoulos later established the nontrivial lower bound $\Delta(n) \geq (\log\log n)^{0.3533227\dots}$ for almost all $n$ , disproving a longstanding conjectured exponent (Ford et al., 2019, Bretèche et al., 2022).

2. Main Theorems: Mean Value and Moment Bounds

Recent advances have substantially tightened the obtainable exponents for both mean value and moments. Koukoulopoulos and Tao established that for all $x \ge 100$ ,

$\sum_{n \le x} \Delta(n) \ll x (\log\log x)^{11/4}$

(Koukoulopoulos et al., 2023), later improved via a refined moment-induction and Fourier smoothing to

$\sum_{n \le x} \Delta(n) \ll x (\log_2 x)^{5/2}$

with a matching lower bound of $x (\log_2 x)^{3/2}$ (Bretèche et al., 2023). Ford, Koukoulopoulos, and Tao pushed the lower bound further to

$\sum_{n \le x} \Delta(n) \gg_\varepsilon x(\log\log x)^{1+\eta-\varepsilon}$

for all $x\geq 100$ and any fixed $\varepsilon>0$ , where $\eta = 0.3533227\ldots$ is the exponent from random set subsum multiplicity (Ford et al., 2023, Ford et al., 2019).

For higher moments, explicit upper bounds have been provided. If $t \ge 1$ , $z > 0$ , and $\varphi \in \mathcal{M}_z$ is a nonnegative multiplicative function satisfying mild conditions, then

$S_{t,\varphi}(x) := \sum_{n \le x} \varphi(n) [\Delta(n)]^t \ll x (\log x)^{3-z} (\log_2 x)^{t+1+O(\varepsilon)}$

if $z \ge t/(2t-1)$ , with variants for lower $z$ and specific results for the quadratic mean $t=2$ (Bretèche et al., 5 Dec 2025). The true order for large $z$ is $S_{t,\varphi}(x) \asymp x (\log x)^{z-1}$ .

3. Methods: Moments, Induction, and Fourier Analysis

A recurring methodology is the analysis of the $q$ -th moment

$M_q(n) = \left( \int_{-\infty}^{\infty} [\Delta(n;u)]^q \, du \right)^{1/q}$

where $\Delta(n;u) = |\{d \mid n : e^u < d \leq e^{u+1}\}|$ , and recursive inequalities for $M_q(n)$ under multiplicative extensions. A key identity is

$\Delta(np;u) = \Delta(n;u) + \Delta(n;u-\log p)$

for $p \nmid n$ , leading to recursive moment bounds via multilinear integrals and combinatorial arguments (Koukoulopoulos et al., 2023, Bretèche et al., 2023). Large-sieve and Parseval (Fourier) techniques are systematically applied to control concentrated divisor "spikes" and to extract probabilistic bounds for $\Delta(n)$ over various domains.

The method of separating $n$ into "smooth" and "rough" parts, combined with probabilistic models emulating the random distribution of prime factors (notably, the logarithmic random set model), underpins the derivation of the strongest lower bounds (Ford et al., 2023, Ford et al., 2019).

4. Distributional Laws, Normal Order, and Explicit Estimates

The normal order of $\Delta(n)$ for typical $n$ is now bracketed between substantial exponents: $(\log\log n)^{0.3533227\dots} \ll \Delta(n) \ll (\log_2 n)^{0.61025}$ for almost all $n$ (Bretèche et al., 2022, Ford et al., 2019). Combinatorial optimization over subset sum multiplicities in random sets, with entropy constraints and probabilistic transference, is central to the lower bound of the normal order (Ford et al., 2019).

In short intervals, Bordelles has obtained fully explicit upper bounds for the sum $\sum_{x-H < n \le x}\Delta(n)$ , tracking all constants: $\sum_{x-y < n \le x} \Delta(n) \leq \phi(\ell)\,y\,(\log x)^{-1+4/\pi}$ where $\phi(\ell)$ is an explicit function of $\ell$ and $y$ varies over $x^{1/\ell} \le y \le x$ (Bordellès, 19 Feb 2024).

5. Generalizations: Twists, Characters, and Number Fields

The function extends to broader contexts: for a Dirichlet character $\chi$ (real or complex), de la Bretèche and Tenenbaum, and later Lartaux (Lartaux, 2020), considered

$\Delta_V(n, \chi) := \sup_{u \in \mathbb{R},\, 0 \leq v \leq V} \left|\sum_{\substack{d\mid n\e^u<d\leq e^{u+v}}} \chi(d)\right|$

and established sharp dependence on the interval parameter $V$ and moment exponents—e.g., $S_{t,V}(x;\chi,g) \ll x\,L(x)^{\alpha} V^t (\log x)^{y-1+W^+}$ , with $L(x)$ and $W$ as defined in the paper (Lartaux, 2020). These results cover complex characters, answer a question of Hooley regarding statistics of local divisor sums in longer intervals, and show that the $V$ dependence is generically $V^t$ for the $2t$-th moment.

Higher degree generalizations appear for $\Delta_3(n,f_1,f_2)$ and its moments, especially with character twists. For instance, Lartaux and de la Bretèche–Tenenbaum proved for nonprincipal characters $\chi_1,\chi_2$ and all $x \geq 16$ ,

$\sum_{n \leq x} \Delta_3(n, \chi_1, \chi_2)^2 \ll x(\log x)^{\rho} \LL(x)^{\alpha}$

where $\rho = \frac{\sqrt{3}}{\pi} - \frac{1}{3} \approx 0.218$ and $\LL(x) = \exp\{\sqrt{\log_2 x\,\log_3 x}\}$ (Lartaux, 2021).

Extensions to arbitrary number fields $K$ with ring of integers $\mathcal{O}_K$ consider

$\Delta_K(\mathfrak{a}) := \sup_{u \in \mathbb{R}} \#\{ \mathfrak{d}\mid\mathfrak{a} : e^u < N\mathfrak{d} \leq e^{u+1}\}$

and establish that $(1/x)\sum_{N\mathfrak{a}\le x} \Delta_K(\mathfrak{a}) \ll (\log x)^{c\,\widehat{\varepsilon}(x)}$ with $\widehat{\varepsilon}(x) = \sqrt{\log\log\log(16+x) / \log\log(3+x)}$ (Sofos, 2016).

6. Friable/Smooth Numbers and Phase Transitions

For friable (smooth) integers with $P^+(n)\leq y$ , the mean value of $\Delta(n)$ exhibits a phase transition depending on the friability parameter $y$ . For $(x,y)$ in suitable regimes, Martin–Tenenbaum–Wetzer proved:

For moderately large $y$ , $S_\Delta(x,y) := \frac{1}{\Psi(x,y)} \sum_{n\in S(x,y)} \Delta(n) \gg \log_2 y + (u)$ , and simultaneously $S_\Delta(x,y) \ll 2^{u+O(u/\log_2 u)} \exp\{c\sqrt{\log_2 y \log_3 y}\}$ .
For very small $y$ , a precise asymptotic $S_\Delta(x,y) = \exp\{g(\lambda)u(1+O(\varepsilon_y + 1/\log_2 u))\}$ holds, with $g(\lambda)$ an explicit function (Martin et al., 2023).

Proofs rely on saddle-point/contour analysis for friable counts, combinatorial convolution with $\tau(n)$ , iterative $q$ -mean methods, and exploitation of Gaussian-like behaviour of $\tau(n)$ among friables.

7. Open Problems and Future Directions

Key remaining open questions include determining the sharp order of $\sum_{n\le x}\Delta(n)$ , narrowing the gap between the best-known lower ( $x(\log\log x)^{1+\eta}$ ) and upper ( $x(\log_2 x)^{5/2}$ ) bounds, and closing the interval for the typical normal order exponent $\beta$ between roughly $0.3533$ and $0.61025$ (Bretèche et al., 2023, Bretèche et al., 2022, Ford et al., 2023, Ford et al., 2019). Optimization in high-multiplicity subset sum problems—combinatorially controlling multiplicities of equal sums in random sets—remains tightly connected to analytic bounds for $\Delta(n)$ . Other major directions concern the second and higher moments over twists, asymptotics for distributions on polynomial values, refinement of explicit short interval bounds, and extension to higher-degree character and number field settings—significant for both foundational theory and applications such as Manin’s conjectures and Diophantine geometry (Sofos, 2016, Lartaux, 2020).