A LOWER BOUND ON THE MEAN VALUE OF THE ERD ˝OS–HOOLEY DELTA FUNCTION

A BSTRACT . We give an improved lower bound for the average of the Erd˝os–Hooley function ∆( n ) , namely P n 6 x ∆( n ) ≫ ε x (log log x ) 1+ η − ε for all x > 100 and any ﬁxed ε , where η = 0 . 3533227 . . . is an exponent previously appearing in work of Green and the ﬁrst two authors. This improves on a previous lower bound of ≫ x log log x of Hall and Tenenbaum, and can be compared to the recent upper bound of x (log log x ) 11 / 4 of the second and third authors.


INTRODUCTION
The Erdős-Hooley Delta function is defined for a natural number n as ∆(n) := max u∈R #{d|n : e u < d e u+1 }.
Erdős introduced this function in the 1970s [4,5] and studied certain aspects of its distribution in joint work with Nicolas [6,7].However, it was not until the work of Hooley in 1979 that ∆ was studied in more detail [15].Specifically, Hooley proved that (1.1) See also Section 2 below for our asymptotic notation conventions.
Hooley's estimate (1.1) has been improved by several authors [11,12,13], [14], [2], [17], culminating in the bounds (1.2) x Log 2 x ≪ n x ∆(n) ≪ x(Log 2 x) 11/4 , with the lower bound established by Hall and Tenenbaum in [11] (see also [14, Theorem 60]), and the upper bound recently established in [17].The main result of the present paper is an improvement of the lower bound in (1.2).Our estimate is given in terms of the best known lower bounds for the normal order of ∆, so we discuss these first.
In [2], La Bretèche and Tenenbaum proved that To state the best known lower bound for the normal order, we need some additional notation, essentially from [10].Definition 1.If A is a finite set of natural numbers, the subsum multiplicity m(A) of A is defined to be the largest number m so that there area distinct subsets A 1 , . . ., A m of A such that (1.3) One can think of the subsum multiplicity as a simplified model for the Erdős-Hooley Delta function.
Now take A to be a random set of natural numbers, in which each natural number a lies in A with an independent probability of P(a ∈ A) = 1/a.If k is a natural number, we define β k to be the supremum of all constants c < 1 such that It is shown in [10], by building on work in [18], that β k exists and is positive for all k.We then define the quantity thus η * is the largest exponent for which one has The main results of [10] can then be summarized as follows: Theorem 1. [10] (i) We have η * η, where η = 0.353327 . . . is defined by the formula and ̺ is the unique number in (0, 1/3) satisfying the equation 1−̺/2 = lim j→∞ 2 j−2 / log a j with a 1 = 2, a 2 = 2 + 2 ̺ and a j = a 2 j−1 + a ̺ j−1 − a 2̺ j−2 for j ∈ Z 3 .(ii) For any ε > 0, one has the lower bound As a consequence of these results, we see that as n → ∞ outside of a set of zero natural density.In particular η η * θ.It is conjectured in [10] that η * = η (and in fact The main purpose of this note is to obtain an analogue of the lower bound in (1.6) for the mean value, thus improving the lower bound in (1.2).
Theorem 2. For any ε > 0 and all x 1, we have Very informally, the idea of proof of the theorem is as follows.Our task is to show that the mean value of ∆(n) is at least (Log 2 x) 1+η * −ε .In our arguments, it will be convenient to use a more "logarithmic" notion of mean in which n is square-free and the prime factors of n behave completely independently; see Section 2 for details.It turns out that for each natural number r in the range there is a significant contribution (of size ≫ ε (Log 2 x) η * −ε ), arising from (squarefree) numbers n whose number, ω(n), of prime factors is precisely r; summing in r will recover the final factor of Log 2 x claimed.Suppose we write r = (1 + α) Log 2 x for some ε α 1 − ε and we define y such that.
It turns out that the dominant contribution to the mean from those numbers with ω(n) = r comes from those n that factor as n = n ′ n ′′ , where n ′ is composed of primes < y, n ′′ is composed of primes y, ω(n A pigeonholing argument (see Lemma 3.1) then gives a lower bound roughly of the form where ∆ * (m) is the maximum number of divisors of m in an interval of the form (e u , ye u ].The two factors τ (n ′ ) log y and ∆ * (n ′′ ) behave independently.The arguments from [10] will allow us to ensure that ∆ * (n ′′ ) ≫ (Log 2 x) η * −ε/2 with high probability, while the constraints on n ′ basically allow us to assert that the τ (n ′ ) log y factor has bounded mean (after summing over all the possible values of ω(n <y )).There is an unwanted loss of about (related to the Erdős-Kac theorem) coming from the restriction to n having exactly r prime factors, but this loss can be recovered by summing over the ≍ Log 2 x essentially distinct possible values of y, after showing some approximate disjointness between events associated to different y (see Lemma 4.1).
Remarks.(a) The above-described behavior that ∆ seems to exhibit is rather unusual.For most arithmetic functions f , there is some constant ̺ > 0 such that the dominant contribution to the partial sums n x f (n) comes from integers n with about r 0 := ̺ Log 2 x prime factors.If we let S r (x) = n x, ω(n)=r f (n), then we often have Gaussian-like decay as r moves away from r 0 , meaning that S r (x) ≈ S r 0 (x)e −c(r−r 0 ) 2 / Log 2 x for some c > 0 (e.g. when f is the k-th divisor function).For some arithmetic functions, we have the even stronger exponential decay S r (x) ≈ S r 0 e −c|r−r 0 | for some c ′ > 0 (e.g. when f is the indicator function of integers with a divisor in a given dyadic interval [y, 2y] -see [9]).However, in the case of f = ∆, the sums S r (x) appear to remain of roughly the same size for all r in the range (1.7); the lower bounds we obtain in Theorem 3 below for S r (x) are of the same order for all r in the range (1.7).
(b) A few months after the publication of an arXiv preprint of the present paper, La Bretèche and Tenenbaum [3] proved the following result: This improves Theorem 2 and the upper bound in (1.2) that was proven in [17].To obtain this improvement, La Bretèche and Tenenbaum refined the methods of [17] and of the present paper.TT is supported by NSF grant DMS-1764034.

NOTATION AND BASIC ESTIMATES
We use X ≪ Y , Y ≫ X, or X = O(Y ) to denote a bound of the form |X| CY for a constant C. If we need this constant to depend on parameters, we indicate this by subscripts, for instance X ≪ k Y denotes a bound of the form |X| C k Y where C k can depend on k.We also write X ≍ Y for X ≪ Y ≪ X.All sums and products will be over natural numbers unless the variable is p, in which case the sum will be over primes.We use ½{E} to denote the indicator of a statement E, thus ½{E} equals 1 when E is true and 0 otherwise.In addition, we write ¬E for the negation of E. We use P for probability and E for probabilistic expectation.
Given an integer n, we write τ (n) := d|n 1 for its divisor-function and ω(n) := p|n 1 for the number of its distinct prime factors.
It will be convenient to work with the following random model of squarefree integers.For each prime p, let n p be a random variable equal to 1 with probability p p+1 , and p with probability 1 p+1 , independently in p. Then for any x 1, define the random natural number In particular we may factor n <x into independent factors n <x = n <y n [y,x) for any 1 y < x.Observe that n <x takes values in the set S <x denoting the set of square-free numbers, all of whose prime factors p are such that p < x, with for all n ∈ S <x .In particular, from Mertens' theorem we have for any non-negative function f : N → R + .We further note that We can also generalize (2.1) to where S [y,x) denotes the set of square-free numbers, all of whose prime factors lie in [y, x).
We have the following elementary inequality: Proposition 2.1.For any x 1, we have Proof.We may take x to be sufficiently large.Restricting attention to numbers n x of the form n = mp where m √ x/2 and √ x < p x/m, we observe that ∆(n) ∆(m), and thus Hence, the Prime Number Theorem [16, Theorem 8.1] implies that Restricting further to those m in S <y , y = x 1/10 , we conclude from (2.1) that ( for large enough x.Combined with (2.4), this concludes the proof.
Thus, to prove Theorem 2, it will suffice (after replacing x with x 1/10 ) to establish the lower bound for all ε > 0, and x sufficiently large in terms of ε.In fact we will show the following stronger estimate.
Theorem 3. Let ε > 0 and x > 0, and let r be an integer in the range Clearly, Theorem 3 implies (2.5) on summing over r.
We first record some basic information (cf.[21], [14, Theorems 08, 09]) about the distribution of ω(n <x ) (or more generally ω(n [y,x) )), reminiscent of the Bennett inequality [1] but with a crucial additional square root gain in the denominator; it can also be thought of as a "large deviations" variant of the Erdős-Kac law.
Proposition 2.2.Let 1 y x, with x sufficiently large, and let k be a positive integer with where t 10.Then we have where Proof.By (2.3) we have , where R y,x := by Mertens' estimate [16, Theorem 3.4(b)] and our assumption that t 10.Using the Stirling approximation k! ≍ k 1/2 (k/e) k , we obtain the claimed upper bound.
Lastly, we prove a corresponding lower bound.Let C be sufficiently large and assume that x e C .Set y 1 = max(y, C), and define By Mertens' estimate, we have and it follows that k 20L.In addition, we have the last inequality holding for large enough C, since y 1 C.By the upper bound on t and inequality (2.7), we have Together with (2.6), and since Log 2 y 1 Log 2 y + Log 2 C, we conclude that Hence, the claimed lower bound on R y,x follows by Stirling's formula We record two particular corollaries of the above proposition of interest, which follow from a routine Taylor expansion of the function Q(t).
, and let k ∈ N. We have the two following cases:

MAIN REDUCTION
Let the notation and hypotheses be as in Theorem 3. We allow all implied constants to depend on ε.We write We may assume x sufficiently large depending on ε.For any 1 y < x, we have To take advantage of the splitting by y, we introduce a generalization of the Erdős-Hooley Delta function for any v > 0, and use the following simple application of the pigeonhole principle.
Lemma 3.1.For any 1 y < x and any v log n <y , we have Proof.By (3.2), there exists u such that there are As we shall see later, both events E y and F y will hold with very high probability.As ∆ (v) is clearly monotone in v, we have By Lemma 3.1 with v = 10(Log 3 x) Log y, if the events E y and F y both hold for some y ∈ Y, then Thus Theorem 3 will follow if we show that (3.5) Controlling the left-hand side is accomplished with the following three propositions.In their statements, recall that ¬G denotes the negation of the event G. Proposition 3.2.We have Proof.We have Proposition 3.3 will be proved in Section 4.
Proposition 3.4 will be proved in Section 5. Now we complete the proof of (3.5), assuming the three propositions above.Firstly, by Proposition 3.4 and the independence of F and n <y , we have Combining this with Propositions 3.2 and 3.3, we deduce (3.5), and hence Theorem 3.

PROOF OF PROPOSITION 3.3
Let y ∈ Y, so that Log 2 y/ log 2 is an integer.Consider the events E y,u := ω(n <x ) = r, ω(n <y ) = Log 2 y log 2 + u .
In the event E y,u we have τ (n <y ) = 2 u Log y.Also, consider the events Thus, In fact, u 2 u−1 P(H u ) u 2 u G u , so we have lost at most a factor 1/2 in the final step.We will restrict attention to the most important values of u, namely u ∈ U, where This choice is informed by the calculations in [17,Proposition 4.1].We then observe that (4.1) It remains to bound P(G u ) from below for u ∈ U.This follows essentially by more general results of Ford [8], but we may give a simple and self-contained argument in the special case we are interested in.To do so, we employ the second moment method.More precisely, we have the following estimates: Lemma 4.1.We have and, for all y, y ′ ∈ Y and u ∈ U, Before we prove Lemma 4.1, let us see how to use it to bound P(G u ) from below.
For any given u ∈ U, Lemma 4.1 yields that On the other hand, the Cauchy-Schwarz inequality implies that Indeed, if µ = m(A), then Definition 1 implies that we can find distinct subsets A 1 , . . ., A µ of A such that Also, for each a ∈ A, we can find a prime p a |n [y,x) such that (5.3) λ a−2 p a < λ a−1 .
In particular, the cardinality of A (and hence of A thanks to the choice of λ.We conclude that all the sums a∈A j log p a , j = 1, . . ., µ lie in an interval of the form (u, u + log y], hence all the products a∈A j p a lie in an interval of the form (e u , e u+log y ].As these products are all distinct factors of n [y,x) , the claim (5.2) follows.
In view of (5.2), it now suffices to establish the bound The events a ∈ A for a ∈ J are independent with a probability of at least 1/a.One can then find a random subset A ′ of A where the events a ∈ A ′ for a ∈ J are independent with a probability of exactly 1/a; for instance, one could randomly eliminate each a ∈ A from A ′ with an independent probability of 1 − 1 aP(a∈A) .Clearly m(A) m(A ′ ), so it will suffice to show that (5.4) Now we use a "tensor power trick" going back to the work of Maier and Tenenbaum [18] (see also [10,Lemma 2.1]).Observe the supermultiplicativity inequality As a consequence of this and (5.6), we can find an integer ℓ satisfying and disjoint sets J 1 , . . ., J ℓ in J, where each J i is of the form J i = [D c i , D i ] ∩ Z for some D i ≫ Log 2 x.From (5.5) and monotonicity, we then have (5.8)m(A ′ ) ℓ i=1 m(A ′ ∩ J i ).
From the definition of η * in (1.4), we have c < β k if k is large enough.From the definition of β k in Definition 1, we conclude (as D i is sufficiently large depending on k, δ) that P m(A ′ ∩ J i ) k 1 − δ for all i = 1, . . ., ℓ.Furthermore, the events m(A ′ ∩ J i ) k are independent, because the sets A ′ ∩ J i are independent.By the Bennett inequality [1], the probability that there are fewer than (1 − ε/5)ℓ values of i with m(A ′ ∩ J i ) k is at most We choose δ small enough so that From another appeal to (5.7) we have The claim (5.4) follows.

4 π − 1 for any x 1 .
Here and in the sequel we use the notation Log x := max{1, log x} for x > 0, and also define Log 2 x := Log(Log x); Log 3 x := Log(Log 2 x); Log 4 x := Log(Log 3 x).

Acknowledgments.
KF is supported by National Science Foundation Grants DMS-1802139 and DMS-2301264.DK is supported by the Courtois Chair II in fundamental research, by the Natural Sciences and Engineering Research Council of Canada (RGPIN-2018-05699) and by the Fonds de recherche du Québec -Nature et technologies (2022-PR-300951).

n
<x := p<x n p and similarly for any 1 y < x define the natural number n [y,x) := y p<x n p .
e u+v ].Multiplying one of these divisors b by any of the τ (n <y ) divisors a of n <y gives a divisor ab of n <x in the range (e u , e u+2v ].These τ (n <y )∆ (v) (n [y,x) ) divisors are all distinct.Covering this range by at most 2v + 1 intervals of the form (e u ′ , e u ′ +1 ], we obtain the claim from the pigeonhole principle.Let (3.3) Y := y > 0 : | Log 2 y − α Log 2 x| Log 2 x ; Log 2 y log 2 ∈ Z and for each y ∈ Y, let E y denote the event log n <y 10(Log 3 x) Log y and let F y denote the event (3.4)
Arguing as in the proof of Proposition 3.2, we have E τ (n <y ) ≪ Log y, and thus E ½{ω(n <x ) = r} max y∈Y τ (n <y ) Log y 1 , . . ., A µ ) is at most ω(n [y,x) ), and hence at most 2 Log 2 x.Taking logarithms in (5.3), we see that | log p a − a log λ| 2 log λ for all a ∈ A. Thus for all 1 j < j ′ µ we have from the triangle inequality, (1.3), and the bound |A| 2 Log 2 x that