---
title: 'Erdős–Hooley Delta-function: Overview'
url: https://www.emergentmind.com/topics/erdos-hooley-delta-function
type: topic
---

# Erdős–Hooley Delta-function: Overview

The Erdős–Hooley Delta-function $\Delta(n)$ is a central object in the analytic study 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 study 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 $\tau(n) \sim x \log x$ [2306.08615]. 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 $\sum_{n\leq x}\Delta(n)\gg x\log\log x$ [2308.11987, 2210.13897].

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 [1908.00378, 2210.13897].

## 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 $x \ge 100$, 
\[
\sum_{n \le x} \Delta(n) \ll x (\log\log x)^{11/4}
\]
[2306.08615], 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 $x (\log_2 x)^{3/2}$ [2309.03958]. 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 [2308.11987, 1908.00378].

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 $t=2$ [2512.05652]. 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 [2306.08615, 2309.03958]. 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 [2308.11987, 1908.00378].

## 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 $n$ [2210.13897, 1908.00378]. 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 [1908.00378].

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 $x^{1/\ell} \le y \le x$ [2402.12333].

## 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 [2003.04209], 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 [2003.04209]. 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 $\LL(x) = \exp\{\sqrt{\log_2 x\,\log_3 x}\}$ [2102.06252].

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 $\widehat{\varepsilon}(x) = \sqrt{\log\log\log(16+x) / \log\log(3+x)}$ [1609.09002].

## 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 [2307.05530].

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 $0.61025$ [2309.03958, 2210.13897, 2308.11987, 1908.00378]. 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 [1609.09002, 2003.04209].

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The recent literature provides a cohesive probabilistic and harmonic-analytic framework for the distribution and moments of the Erdős–Hooley Delta-function, connecting analytic number theory with combinatorial optimization and probabilistic models of divisor structures [2210.13897, 2306.08615, 2308.11987, 2309.03958, 2512.05652, 2003.04209, 2102.06252, 1609.09002, 2402.12333, 2307.05530, 1908.00378].

Source: https://www.emergentmind.com/topics/erdos-hooley-delta-function