A decrease sure on the imply worth of the Erd”os-Hooley delta perform

Math

A decrease sure on the imply worth of the Erd”os-Hooley delta perform

keiseronlineuniversity.com

24 August 2023

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Kevin Ford, Dimitris Koukoulopolous and I’ve simply uploaded to the arXiv our paper “A decrease sure on the imply worth of the Erdös-Hooley delta perform“. This paper enhances the current paper of Dimitris and myself acquiring the higher sure

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

on the imply worth of the Erdös-Hooley delta perform

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

On this paper we acquire a decrease sure

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

the place ${eta = 0.3533227dots}$ is an exponent that arose in earlier work of results of Ford, Inexperienced, and Koukoulopolous, who confirmed that

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

for all ${n}$ exterior of a set of density zero. The earlier finest recognized decrease sure for the imply worth was

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

because of Corridor and Tenenbaum.

The purpose is the primary contributions to the imply worth of ${Delta(n)}$ are pushed not by “typical” numbers ${n}$ of some dimension ${x}$ , however reasonably of numbers which have a splitting

$displaystyle n = n' n''$

the place ${n''}$ is the product of primes between some intermediate threshold ${1 leq y leq x}$ and ${x}$ and behaves “sometimes” (so specifically, it has about ${loglog x - loglog y + O(sqrt{loglog x})}$ prime components, as per the Hardy-Ramanujan legislation and the Erdös-Kac legislation, however ${n'}$ is the product of primes as much as ${y}$ and has double the variety of typical prime components – ${2 loglog y + O(sqrt{loglog x})}$ , reasonably than ${loglog y + O(sqrt{loglog x})}$ – thus ${n''}$ is the kind of quantity that might make a major contribution to the imply worth of the divisor perform ${tau(n'')}$ . Right here ${y}$ is such that ${loglog y}$ is an integer within the vary

$displaystyle varepsilonloglog x leq log log y leq (1-varepsilon) loglog x$

for some small fixed ${varepsilon>0}$ there are mainly ${loglog x}$ completely different values of ${y}$ give primarily disjoint contributions. From the simple inequalities

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

(the latter coming from the pigeonhole precept) and the truth that ${frac{tau(n')}{log n'}}$ has imply about one, one would count on to get the above end result supplied that one may get a decrease sure of the shape

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

for most common ${n''}$ with prime components between ${y}$ and ${x}$ . Sadly, because of the lack of small prime components in ${n''}$ , the arguments of Ford, Inexperienced, Koukoulopoulos that give (1) for typical ${n}$ don’t fairly work for the rougher numbers ${n''}$ . Nevertheless, it seems that one can get round this drawback by changing (2) by the extra environment friendly inequality

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

the place

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

is an enlarged model of ${Delta^{(n)}}$ when ${v geq 1}$ . This inequality is definitely confirmed by making use of the pigeonhole precept to the components of ${n}$ of the shape ${d' d''}$ , the place ${d'}$ is likely one of the ${tau(n')}$ components of ${n'}$ , and ${d''}$ is likely one of the ${Delta^{(log n')}(n'')}$ components of ${n''}$ within the optimum interval ${[e^u, e^{u+log n'}]}$ . The additional room supplied by the enlargement of the vary ${[e^u, e^{u+1}]}$ to ${[e^u, e^{u+log n'}]}$ seems to be adequate to adapt the Ford-Inexperienced-Koukoulopoulos argument to the tough setting. The truth is we’re in a position to make use of the primary technical estimate from that paper as a “black field”, particularly that if one considers a random subset ${A}$ of ${[D^c, D]}$ for some small ${c>0}$ and sufficiently giant ${D}$ with every ${n in [D^c, D]}$ mendacity in ${A}$ with an impartial chance ${1/n}$ , then with excessive chance there ought to be ${gg c^{-1/eta+o(1)}}$ subset sums of ${A}$ that attain the identical worth. (Initially, what “excessive chance” means is simply “near ${1}$ “, however one can scale back the failure chance considerably as ${c rightarrow 0}$ by a “tensor energy trick” profiting from Bennett’s inequality.)

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