A observe on the imply worth of the Hooley delta perform

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A observe on the imply worth of the Hooley delta perform

keiseronlineuniversity.com

12 August 2023

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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

$displaystyle Delta(n) := sup_u Delta(n;u)$

the place

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

The perform ${Delta}$ measures the extent to which the divisors of a pure quantity could be concentrated in a dyadic (or extra exactly, ${e}$ -dyadic) interval ${(e^u, e^{u+1}]}$ . From the pigeonhole precept, we have now the bounds

$displaystyle frac{tau(n)}{log n} ll Delta(n) leq tau(n),$

the place ${tau(n) := # n}$ is the standard divisor perform. The statistical conduct of the divisor perform is nicely understood; as an illustration, if ${n}$ is drawn at random from ${1}$ to ${x}$ , then the imply worth of ${tau(n)}$ is roughly ${log x}$ , the median is roughly ${log^{log 2} x}$ , and (by the Erdos-Kac theorem) ${tau(n)}$ 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 ${n}$ at random from ${1}$ to ${x}$ for giant ${x}$ , the median is thought to be someplace between ${(loglog x)^{0.3533dots}}$ and ${(loglog x)^{0.6102dots}}$ for giant ${x}$ – 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 ${frac{1}{x} sum_{n leq x} Delta(n)}$ was even much less nicely managed; the most effective earlier bounds have been

$displaystyle log log x ll frac{1}{x} sum_{n leq x} Delta(n) ll exp( c sqrt{loglog x} )$

for any ${c > sqrt{2} log 2}$ , with the decrease certain as a result 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

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

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 ${A}$ of ${{1,dots,N}}$ with the property that no non-empty subset of ${A}$ sums to an ideal sq.? Erdos noticed that one can acquire units of dimension ${gg N^{1/3}}$ (mainly by contemplating sure homogeneous arithmetic progressions), and Nguyen and Vu confirmed an higher certain of ${ll N^{1/3} (log N)^{O(1)}}$ . 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 ${ll N^{1/3} (loglog N)^{O(1)})}$ .

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 ${n}$ components as ${n = d^2 m}$ with ${m}$ squarefree, then ${Delta(n) leq tau(d^2) Delta(m)}$ ). Subsequent, as a result of a square-free quantity ${n>1}$ could be uniquely factored as ${n = pm}$ the place ${p}$ is a main and ${m}$ lies within the finite set ${{mathcal S}_{<p}}$ of squarefree numbers whose prime components are lower than ${p}$ , and ${Delta(n) leq tau(p) Delta(m) = 2 Delta(m)}$ , it isn’t tough to ascertain the certain

$displaystyle frac{1}{x} sum_{n in {mathcal S}_{<x}} Delta(n) ll sup_{2 leq yleq x} frac{1}{log y} sum_{n in {mathcal S}_{<y}} frac{Delta(n)}{n}.$

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

$displaystyle frac{1}{log x} sum_{n in {mathcal S}_{<x}} frac{Delta(n)}{n} ll (log log x)^{11/4}. (1)$

We really show a barely extra refined distributional estimate: for any ${A geq 2}$ , we have now a certain

$displaystyle Delta(n) ll A log^{3/4} A (2)$

exterior of an distinctive set ${E}$ which is small within the sense that

$displaystyle frac{1}{log x} sum_{n in {mathcal S}_{<x} x: n in E} frac{1}{n} ll frac{1}{A}. (3)$

It’s not tough to get from this distributional estimate to the logarithmic common estimate (1) (worsening the exponent ${3/4}$ to ${3/4+2 = 11/4}$ ).

To get some instinct on the scale of ${Delta(n)}$ , we observe that if ${y > 0}$ and ${n_{<y}}$ is the issue of ${n}$ coming from the prime components lower than ${y}$ , then

$displaystyle Delta(n) geq Delta(n_{<y}) gg frac{tau(n_{<y})}{log n_{<y}}. (4)$

Alternatively, commonplace estimates let one set up that

$displaystyle tau(n_{<y}) ll A log n_{<y} (5)$

for all ${y}$ , and all ${n}$ 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 ${log y}$ is near ${A^{log 4}}$ , 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 ${Delta(n) = sup_u Delta(n;u)}$ by the moments

$displaystyle M_q(n) := int_{{bf R}} Delta(n,u)^q du$

for pure numbers ${q}$ ; it isn’t tough to ascertain the certain

$displaystyle Delta(n) ll M_q(n)^{1/q}$

and one expects this certain to change into basically sharp as soon as ${q sim loglog x}$ . We will present a second certain

$displaystyle sum_{n in {mathcal S}_{<x} backslash E_q} frac{M_q(n) / tau(n)}{n} leq O(q)^q A^{q-2} log^{3q/4} A$

for any ${q geq 2}$ for some distinctive set ${E_q}$ obeying the smallness situation (3) (really, for technical causes we have to enhance the right-hand aspect barely to shut an induction on ${q}$ ); it will indicate the distributional certain (2) from a normal Markov inequality argument (setting ${q sim loglog x}$ ).

The technique is then to acquire recursive inequality for (averages of) ${M_q(n)}$ . As within the discount to (1), we issue ${n=pm}$ the place ${p}$ is a main and ${m in {mathcal S}_{<p}}$ . One observes the identification

$displaystyle Delta(n;u) = Delta(m;u) + Delta(m;u-log p)$

for any ${u}$ ; taking moments, one obtains the identification

$displaystyle M_q(n) = sum_{a+b=q; 0 leq b leq q} binom{q}{a} int_{bf R} Delta(m;u)^a Delta(m;u-log p)^b du.$

As in earlier literature, one can attempt to common in ${p}$ right here and apply Hölder’s inequality. But it surely handy to first use the symmetry of the summand in ${a,b}$ to scale back to the case of comparatively small values of ${b}$ :

$displaystyle M_q(n) leq 2 sum_{a+b=q; 0 leq b leq q/2} binom{q}{a} int_{bf R} Delta(m;u)^a Delta(m;u-log p)^b du.$

One can extract out the ${b=0}$ time period as

$displaystyle M_q(n) leq 2 M_q(m)$

$displaystyle + 2 sum_{a+b=q; 1 leq b leq q/2} binom{q}{a} int_{bf R} Delta(m;u)^a Delta(m;u-log p)^b du.$

It’s handy to get rid of the issue of ${2}$ by dividing out by the divisor perform:

$displaystyle frac{M_q(n)}{tau(n)} leq frac{M_q(m)}{tau(m)}$

$displaystyle + frac{1}{m} sum_{a+b=q; 1 leq b leq q/2} binom{q}{a} int_{bf R} Delta(m;u)^a Delta(m;u-log p)^b du.$

This inequality is appropriate for iterating and likewise averaging in ${p}$ and ${m}$ . After some commonplace manipulations (utilizing the Brun–Titchmarsh and Hölder inequalities), one is ready to estimate sums similar to

$displaystyle sum_{n in {mathcal S}_{<x} backslash E_q} frac{M_q(n)/tau(n)}{n} (6)$

by way of sums similar to

$displaystyle int_2^{x^2} sum_{a+b=q; 1 leq b leq q/2} binom{q}{a} sum_{n in {mathcal S}_{<x} backslash E_q} frac{M_a(n) M_b(n)}{tau(n) n} frac{dy}{log^2 y}$

(assuming a sure monotonicity property of the distinctive set ${E_q}$ 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 ${M_b}$ (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

$displaystyle sum_{n in {mathcal S}_{<x} backslash E_a} frac{M_a(n)/tau(n)}{n}$

for varied ${a < q}$ . This permits one to estimate these expressions effectively by induction.

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