# A word on the imply worth of the Hooley delta operate

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Dimitris Koukoulopolous and I’ve simply uploaded to the arXiv our paper “A word on the imply worth of the Hooley delta operate“. This paper issues a (nonetheless considerably poorly understood) fundamental arithmetic operate in multiplicative quantity concept, specifically the *Hooley delta operate*

the place

The operate measures the extent to which the divisors of a pure quantity may be concentrated in a dyadic (or extra exactly, -dyadic) interval . From the pigeonhole precept, we’ve the bounds

the place is the standard divisor operate. The statistical habits of the divisor operate is properly understood; as an example, if is drawn at random from to , then the imply worth of is roughly , the median is roughly , and (by the Erdos-Kac theorem asymptotically has a log-normal distribution). Particularly, there are a small proportion of extremely divisible numbers that skew the imply to be considerably greater than the median.

However, the statistical habits of the Hooley delta operate is considerably much less properly understood, even conjecturally. Once more drawing at random from to for big , the median is thought to be someplace between and for big – a (tough) current results of Ford, Inexperienced, and Koukoulopolous (for the decrease certain) and de la Bréteche and Tenenbaum (for the higher certain). And the imply was even much less properly managed; the most effective earlier bounds have been

for any , with the decrease certain attributable to Corridor and Tenenbaum, and the higher certain a current results of de la Bréteche and Tenenbaum.

The primary results of this paper is an enchancment of the higher certain to

It’s nonetheless unclear to us precisely what to conjecture relating to the precise order of the imply worth.

The rationale we regarded into this drawback was that it was linked to forthcoming work of David Conlon, Jacob Fox, and Huy Pham on the next drawback of Erdos: what’s the dimension of the biggest subset of with the property that no non-empty subset of sums to an ideal sq.? Erdos noticed that one can get hold of units of dimension (principally by contemplating sure homogeneous arithmetic progressions), and Nguyen and Vu confirmed an higher certain of . With our imply worth certain as enter, along with a number of new arguments, Conlon, Fox, and Pham have been capable of enhance the higher certain to .

Let me now focus on among the elements of the proof. The primary few steps are normal. Firstly we might prohibit consideration to square-free numbers with out a lot problem (the purpose being that if a quantity components as with squarefree, then ). Subsequent, as a result of a square-free quantity may be uniquely factored as the place is a primary and lies within the finite set of squarefree numbers whose prime components are lower than , and , it’s not tough to ascertain the certain

The upshot of that is that one can change an extraordinary common with a logarithmic common, thus it suffices to indicate

We really show a barely extra refined distributional estimate: for any , we’ve a certain

outdoors of an distinctive set which is small within the sense that

It’s not tough to get from this distributional estimate to the logarithmic common estimate (1) (worsening the exponent to ).

To get some instinct on the dimensions of , we observe that if and is the issue of coming from the prime components lower than , then

However, normal estimates let one set up that

for all , and all outdoors of an distinctive set that’s small within the sense (3); the truth is it seems that one can even get a further acquire on this estimate until is near , which seems to be helpful when optimizing the bounds. So we wish to roughly reverse the inequalities in (4) and get from (5) to (2), probably after throwing away additional distinctive units of dimension (3).

At this level we carry out one other normal approach, specifically the second methodology of controlling the supremum by the moments

for pure numbers ; it’s not tough to ascertain the certain

and one expects this certain to turn into primarily sharp as soon as . We will present a second certain

for any for some distinctive set obeying the smallness situation (3) (really, for technical causes we have to enhance the right-hand aspect barely to shut an induction on ); it will indicate the distributional certain (2) from an ordinary Markov inequality argument (setting ).

The technique is then to acquire recursive inequality for (averages of) . As within the discount to (1), we issue the place is a primary and . One observes the identification

for any ; taking moments, one obtains the identification

As in earlier literature, one can attempt to common in right here and apply Hölder’s inequality. Nevertheless it handy to first use the symmetry of the summand in to scale back to the case of comparatively small values of :

One can extract out the time period as

It’s handy to get rid of the issue of by dividing out by the divisor operate:

This inequality is appropriate for iterating and in addition averaging in and . After some normal manipulations (utilizing the Brun–Titchmarsh and Hölder inequalities), one is ready to estimate sums resembling

when it comes to sums resembling

(assuming a sure monotonicity property of the distinctive set that seems to carry in our software). By an induction speculation and a Markov inequality argument, one can get an inexpensive pointwise higher certain on (after eradicating one other distinctive set), and the online result’s that one can principally management the sum (6) when it comes to expressions resembling

for varied . This enables one to estimate these expressions effectively by induction.

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