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Dimitris Koukoulopoulos and I’ve simply uploaded to the arXiv our paper “A notice on the imply worth of the Hooley delta perform“. This paper issues a (nonetheless considerably poorly understood) fundamental arithmetic perform in multiplicative quantity idea, particularly the *Hooley delta perform*

the place

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

the place is the same old divisor perform. The statistical conduct of the divisor perform is properly understood; as an illustration, 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.

Alternatively, the statistical conduct of the Hooley delta perform is considerably much less properly understood, even conjecturally. Once more drawing at random from to for giant , the median is thought to be someplace between and for giant – a (troublesome) current results of Ford, Inexperienced, and Koukoulopolous (for the decrease certain) and La Bretèche and Tenenbaum (for the higher certain). And the imply was even much less properly managed; one of the best earlier bounds had been

for any , with the decrease certain on account of Corridor and Tenenbaum, and the higher certain a current results of La Bretèche and Tenenbaum.

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

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

The rationale we seemed into this downside was that it was related to forthcoming work of David Conlon, Jacob Fox, and Huy Pham on the next downside of Erdos: what’s the dimension of the most important subset of with the property that no non-empty subset of sums to an ideal sq.? Erdos noticed that one can receive 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 in a position to enhance the higher certain to .

Let me now focus on a number of the components 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 could be uniquely factored as the place is a chief and lies within the finite set of squarefree numbers whose prime components are lower than , and , it’s not troublesome to determine the certain

The upshot of that is that one can exchange an abnormal common with a logarithmic common, thus it suffices to point out

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

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

It isn’t troublesome to get from this distributional estimate to the logarithmic common estimate (1) (worsening the exponent to ).

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

Alternatively, 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 also can get an extra achieve 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 method, particularly the second methodology of controlling the supremum by the moments

for pure numbers ; it’s not troublesome to determine the certain

and one expects this certain to turn out to be basically sharp as soon as . We can 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 ); this may suggest 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 chief 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. But it surely handy to first use the symmetry of the summand in to cut back to the case of comparatively small values of :

One can extract out the time period as

It’s handy to eradicate the issue of by dividing out by the divisor perform:

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 similar to

when it comes to sums similar to

(assuming a sure monotonicity property of the distinctive set that seems to carry in our utility). 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 similar to

for numerous . This permits one to estimate these expressions effectively by induction.

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