Mathematics
Mathematics is the study of numbers, shapes, and patterns.
Mathematics is the study of numbers, shapes, and patterns.
Organizer:
- Organized by
-
-
A note on the mean value of the Hooley delta function
-
A note on the mean value of the Hooley delta function
Dimitris Koukoulopolous and I have just uploaded to the arXiv our paper “A note on the mean value of the Hooley delta function“. This paper concerns a (still somewhat poorly understood) basic arithmetic function in multiplicative number theory, namely the Hooley delta function
where
The function measures the extent to which the divisors of a natural number can be concentrated in a dyadic (or more precisely, -dyadic) interval . From the pigeonhole principle, we have the bounds
where is the usual divisor function. The statistical behavior of the divisor function is well understood; for instance, if is drawn at random from to , then the mean value of is roughly , the median is roughly , and (by the Erdos-Kac theorem asymptotically has a log-normal distribution). In particular, there are a small proportion of highly divisible numbers that skew the mean to be significantly higher than the median.
On the other hand, the statistical behavior of the Hooley delta function is significantly less well understood, even conjecturally. Again drawing at random from to for large , the median is known to be somewhere between and for large – a (difficult) recent result of Ford, Green, and Koukoulopolous. And the mean was even less well controlled; the best previous bounds were
for any , with the lower bound due to Hall and Tenenbaum, and the upper bound a recent result of de la Bréteche and Tenenbaum.
The main result of this paper is an improvement of the upper bound to
It is still unclear to us exactly what to conjecture regarding the actual order of the mean value is.
The reason we looked into this problem was that it was connected to forthcoming work of David Conlon, Jacob Fox, and Huy Pham on the following problem of Erdos: what is the size of the largest subset of with the property that no non-empty subset of sums to a perfect square? Erdos observed that one can obtain sets of size (basically by considering certain homogeneous arithmetic progressions), and Nguyen and Vu showed an upper bound of . With our mean value bound as input, together with several new arguments, Conlon, Fox, and Pham have been able to improve the upper bound to .
Let me now discuss some of the ingredients of the proof. The first few steps are standard. Firstly we may restrict attention to square-free numbers without much difficulty (the point being that if a number factors as with squarefree, then ). Next, because a square-free number can be uniquely factored as where is a prime and lies in the finite set of squarefree numbers whose prime factors are less than , and , it is not difficult to establish the bound
The upshot of this is that one can replace an ordinary average with a logarithmic average, thus it suffices to show
We actually prove a slightly more refined distributional estimate: for any , we have a bound
outside of an exceptional set which is small in the sense that
It is not difficult to get from this distributional estimate to the logarithmic average estimate (1) (worsening the exponent to ).
To get some intuition on the size of , we observe that if and is the factor of coming from the prime factors less than , then
On the other hand, standard estimates let one establish that
for all , and all outside of an exceptional set that is small in the sense (3); in fact it turns out that one can also get an additional gain in this estimate unless is close to , which turns out to be useful when optimizing the bounds. So we would like to approximately reverse the inequalities in (4) and get from (5) to (2), possibly after throwing away further exceptional sets of size (3).
At this point we perform another standard technique, namely the moment method of controlling the supremum by the moments
for natural numbers ; it is not difficult to establish the bound
and one expects this bound to become essentially sharp once . We will be able to show a moment bound
for any for some exceptional set obeying the smallness condition (3) (actually, for technical reasons we need to improve the right-hand side slightly to close an induction on ); this will imply the distributional bound (2) from a standard Markov inequality argument (setting ).
The strategy is then to obtain a good recursive inequality for (averages of) . As in the reduction to (1), we factor where is a prime and . One observes the identity
for any ; taking moments, one obtains the identity
As in previous literature, one can try to average in here and apply Hölder’s inequality. But it convenient to first use the symmetry of the summand in to reduce to the case of relatively small values of :
One can extract out the term as
It is convenient to eliminate the factor of by dividing out by the divisor function:
This inequality is suitable for iterating and also averaging in and . After some standard manipulations (using the Brun–Titchmarsh and Hölder inequalities), one is able to estimate sums such as
in terms of sums such as
(assuming a certain monotonicity property of the exceptional set that turns out to hold in our application). By an induction hypothesis and a Markov inequality argument, one can get a reasonable pointwise upper bound on (after removing another exceptional set), and the net result is that one can basically control the sum (6) in terms of expressions such as
for various . This allows one to estimate these expressions efficiently by induction.
What’s new • Read More
-
0 Replies
Sorry, there were no replies found.
Log in to reply.