A lower bound on the mean value of the Erdős-Hooley delta function

What's new 2023-08-24

Kevin Ford, Dimitris Koukoulopoulos and I have just uploaded to the arXiv our paper “A lower bound on the mean value of the Erdős-Hooley delta function“. This paper complements the recent paper of Dimitris and myself obtaining the upper bound

$\displaystyle \frac{1}{x} \sum_{n \leq x} \Delta(n) \ll (\log\log x)^{11/4}$

on the mean value of the Erdős-Hooley delta function

$\displaystyle \Delta(n) := \sup_u \# \{ d|n: e^u < d \leq e^{u+1} \}$

In this paper we obtain a lower bound

$\displaystyle \frac{1}{x} \sum_{n \leq x} \Delta(n) \gg (\log\log x)^{1+\eta-o(1)}$

where ${\eta = 0.3533227\dots}$ is an exponent that arose in previous work of result of Ford, Green, and Koukoulopoulos, who showed that

$\displaystyle \Delta(n) \gg (\log\log n)^{\eta-o(1)} \ \ \ \ \ (1)$

for all ${n}$ outside of a set of density zero. The previous best known lower bound for the mean value was

$\displaystyle \frac{1}{x} \sum_{n \leq x} \Delta(n) \gg \log\log x,$

due to Hall and Tenenbaum.

The point is the main contributions to the mean value of ${\Delta(n)}$ are driven not by “typical” numbers ${n}$ of some size ${x}$ , but rather of numbers that have a splitting

$\displaystyle n = n' n''$

where ${n''}$ is the product of primes between some intermediate threshold ${1 \leq y \leq x}$ and ${x}$ and behaves “typically” (so in particular, it has about ${\log\log x - \log\log y + O(\sqrt{\log\log x})}$ prime factors, as per the Hardy-Ramanujan law and the Erdős-Kac law, but ${n'}$ is the product of primes up to ${y}$ and has double the number of typical prime factors – ${2 \log\log y + O(\sqrt{\log\log x})}$ , rather than ${\log\log y + O(\sqrt{\log\log x})}$ – thus ${n''}$ is the type of number that would make a significant contribution to the mean value of the divisor function ${\tau(n'')}$ . Here ${y}$ is such that ${\log\log y}$ is an integer in the range

$\displaystyle \varepsilon\log\log x \leq \log \log y \leq (1-\varepsilon) \log\log x$

for some small constant ${\varepsilon>0}$ there are basically ${\log\log x}$ different values of ${y}$ give essentially disjoint contributions. From the easy inequalities

$\displaystyle \Delta(n) \gg \Delta(n') \Delta(n'') \geq \frac{\tau(n')}{\log n'} \Delta(n'') \ \ \ \ \ (2)$

(the latter coming from the pigeonhole principle) and the fact that ${\frac{\tau(n')}{\log n'}}$ has mean about one, one would expect to get the above result provided that one could get a lower bound of the form

$\displaystyle \Delta(n'') \gg (\log \log n'')^{\eta-o(1)} \ \ \ \ \ (3)$

for most typical ${n''}$ with prime factors between ${y}$ and ${x}$ . Unfortunately, due to the lack of small prime factors in ${n''}$ , the arguments of Ford, Green, Koukoulopoulos that give (1) for typical ${n}$ do not quite work for the rougher numbers ${n''}$ . However, it turns out that one can get around this problem by replacing (2) by the more efficient inequality

$\displaystyle \Delta(n) \gg \frac{\tau(n')}{\log n'} \Delta^{(\log n')}(n'')$

where

$\displaystyle \Delta^{(v)}(n) := \sup_u \# \{ d|n: e^u < d \leq e^{u+v} \}$

is an enlarged version of ${\Delta^{(n)}}$ when ${v \geq 1}$ . This inequality is easily proven by applying the pigeonhole principle to the factors of ${n}$ of the form ${d' d''}$ , where ${d'}$ is one of the ${\tau(n')}$ factors of ${n'}$ , and ${d''}$ is one of the ${\Delta^{(\log n')}(n'')}$ factors of ${n''}$ in the optimal interval ${[e^u, e^{u+\log n'}]}$ . The extra room provided by the enlargement of the range ${[e^u, e^{u+1}]}$ to ${[e^u, e^{u+\log n'}]}$ turns out to be sufficient to adapt the Ford-Green-Koukoulopoulos argument to the rough setting. In fact we are able to use the main technical estimate from that paper as a “black box”, namely that if one considers a random subset ${A}$ of ${[D^c, D]}$ for some small ${c>0}$ and sufficiently large ${D}$ with each ${n \in [D^c, D]}$ lying in ${A}$ with an independent probability ${1/n}$ , then with high probability there should be ${\gg c^{-1/\eta+o(1)}}$ subset sums of ${A}$ that attain the same value. (Initially, what “high probability” means is just “close to ${1}$ “, but one can reduce the failure probability significantly as ${c \rightarrow 0}$ by a “tensor power trick” taking advantage of Bennett’s inequality.)