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.)