Dimitris Koukoulopoulos and I’ve simply uploaded to the arXiv our paper “A observe on the imply worth of the Hooley delta perform“. This paper considerations a (nonetheless considerably poorly understood) fundamental arithmetic perform in multiplicative quantity concept, particularly the Hooley delta perform
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 have now the bounds
the place is the standard divisor perform. The statistical conduct of the divisor perform is nicely 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. Specifically, there are a small proportion of extremely divisible numbers that skew the imply to be considerably increased than the median.
Alternatively, the statistical conduct of the Hooley delta perform is considerably much less nicely understood, even conjecturally. Once more drawing at random from to for giant , the median is thought to be someplace between and for giant – a (tough) 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 nicely managed; the most effective earlier bounds have been
The principle 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 seemed 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 acquire units of dimension (mainly 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 talk about a number of the components of the proof. The primary few steps are commonplace. Firstly we could limit consideration to square-free numbers with out a lot issue (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 main and lies within the finite set of squarefree numbers whose prime components are lower than , and , it isn’t tough to ascertain the certain
for all , and all exterior of an distinctive set that’s small within the sense (3); in actual fact it seems that one may also get a further acquire on this estimate until is near , which seems to be helpful when optimizing the bounds. So we want 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 commonplace method, particularly the second technique of controlling the supremum by the moments
for pure numbers ; it isn’t tough to ascertain the certain
and one expects this certain to change into basically 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 a normal 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 main 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 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 perform:
This inequality is appropriate for iterating and likewise averaging in and . After some commonplace manipulations (utilizing the Brun–Titchmarsh and Hölder inequalities), one is ready to estimate sums similar toby way of 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 mainly management the sum (6) by way of expressions similar to
for varied . This permits one to estimate these expressions effectively by induction.