Squarefrees are Gaussian in short intervals

1.1 Statistics of counts of squarefrees

Let S ⊂ ℕ be the set of squarefree natural numbers (that is natural numbers without a repeated prime factor; by convention, we include 1 ∈ S ). We write

for the number of squarefrees no more than x. It is well known that N S ⁢ ( x ) ∼ x / ζ ⁢ ( 2 ) ; thus the squarefrees have asymptotic density 1 / ζ ⁢ ( 2 ) = 6 / π 2 . Our purpose in this note is to investigate their distribution at a finer scale. In particular, we will investigate the distribution of squarefrees in a random interval ( n , n + H ] , where n is an integer chosen uniformly at random from 1 to X, with X → ∞ .

If H is fixed and does not grow with X, at most O ⁢ ( 1 ) squarefrees can lie in such an interval. Their distribution as X → ∞ is slightly complicated but completely understood; it may be described by Hardy–Littlewood type correlations which can be derived from elementary sieve theory (see [32, 31]). Or, more abstractly, the distribution of squarefrees in an interval of size O ⁢ ( 1 ) can be described by a non-weakly mixing stationary ergodic process (see [7, 43]).

For H tending to infinity with X, matters become at once simpler and more difficult; simpler because some of the irregularities in the distribution just described are smoothed out at this scale, but more difficult in that natural conjectures become more difficult to prove.

Let

be the count of squarefrees in the interval ( n , n + H ] . R. R. Hall [16, 17] was the first to investigate the distribution of this count when H grows with X. In [16, Corollary 1], Hall proved that the variance of the number of squarefrees is of order H if H is not too large with respect to X. More exactly, as X → ∞ , we have

as long as H → ∞ with H ≤ X 2 / 9 - ε .

Keating and Rudnick [25] studied this problem in a function field setting, connecting it with Random Matrix Theory, and suggested based on this that (1.1) will hold for H ≤ X 1 - ε . The best known result is [13], where it is shown that (1.1) holds for H ≤ X 6 / 11 - ε unconditionally and H ≤ X 2 / 3 - ε on the Lindelöf Hypothesis. In fact, in [13], it is shown that even an upper bound of order H for H ≤ X 1 - ε for all ε > 0 would already imply the Riemann Hypothesis.

Because N S ⁢ ( n , H ) is on average of order H, one might naively have expected the variance to also be of order H. That the variance is of order H speaks to the fact that the squarefrees are a rather rigid sequence. This can be discerned even visually in comparison, for instance, to the primes (Figure 1), and we will return to give a more exact description of it in Section 1.3.

In [17], Hall^[1] studied higher moments of counts of squarefrees in short intervals

where k is a positive integer, proving the upper bound lim X → ∞ ⁡ M k ⁢ ( X , H ) ≪ k H ( k - 1 ) / 2 . Various authors have asked whether this can be refined, with the most recent result,

for any ε > 0 as long as H ≤ X 4 / ( 9 ⁢ k ) - ε , being due to Nunes [37]. For H in the range considered, this is an optimal upper bound up to the factor of H ε . In [3], extensive numerical evidence is presented that suggests that these moments are in fact Gaussian.

Our first main result confirms this conjecture.

Thus if H → ∞ and H ≤ X 4 / ( 9 ⁢ k ) - ε for some ε > 0 , we have

Note that the main term is the k-th moment of a centered Gaussian random variable with variance A ⁢ H .

If H = X o ⁢ ( 1 ) , then for any fixed k, we have that H satisfies H ≤ X 4 / ( 9 ⁢ k ) - ε for some ε > 0 for sufficiently large X. Hence, by the moment method (see [4, Section 30]), we obtain the following result.

That is, the centered, normalized counts tend in distribution to a Gaussian random variable.

Gaussian limit theorems are known for the sums over short intervals of several important arithmetic functions (for instance divisor functions d k (see [28]) and the sums-of-squares representation function r (see [20])), and Gaussian limit theorems for counts of primes in short intervals are known under the assumption of strong versions of the Hardy–Littlewood conjectures [33], but Theorem 1.2 seems to be the first instance of an unconditional proof of Gaussian behavior for short interval counts of a non-trivial, natural number theoretic sequence.

Figure 1

A comparison between a short interval containing 125 primes with a short interval containing 125 squarefrees. The relative paucity of gaps and clusters of squarefrees is indicative of the rigidity of their distribution.

(a)

A short interval near 100000 containing 125 primes.

(b)

A short interval near 100000 containing 125 squarefrees.

Hall in [17] asked also about the order of magnitude of the absolute moments

and as a standard corollary of Theorems 1.1 and 1.2, we obtain an asymptotic formula for these.

1.2 B-frees

It is natural to write our proofs in the more general setting of B-free numbers. We recall their definition shortly, but first we fix some notation.

For a sequence J of natural numbers, we will write 𝟏 J for the indicator function of J, and

for the count of elements of J no more than x.

For instance, the function L ⁢ ( x ) = ( log ⁡ x ) j is slowly varying for any j ∈ ℝ . For any slowly varying function L, it is necessary that L ⁢ ( x ) = x o ⁢ ( 1 ) , but this condition is not sufficient. Clearly, in the definition above, α will be the index of the regularly varying sequence J. Further information on regularly varying sequences can be found in [41, Chapter 4.1].

Fix a non-empty subset B ⊂ ℕ > 1 of pairwise coprime integers with ∑ b ∈ B 1 / b < ∞ ; we call such a set a sieving set. We say that a positive integer is B-free if it is indivisible by every element of B. For instance, if B = { p 2 : p ⁢ prime } , B-frees are nothing but squarefrees. Another studied example is B = { p m : p ⁢ prime } for some m ≥ 3 , for which B-frees are the m-th-power free numbers, i.e., integers indivisible by an m-th power of a prime. The notion of B-frees was introduced by Erdős [12], who was motivated by Roth’s work [42] on gaps between squarefrees. (See also [6, 22] for the closely related notion of convergent sieves. We also mention that [22] upper bounded the quantity M ~ 2 ⁢ ( x ; q ) mentioned in Remark 1.4.)

We write 𝟏 B ⁢ -free : ℕ → ℂ for the indicator function of B-free integers, and N B ⁢ -free ⁢ ( x ) for the number of B-free integers n ≤ x . For the sets B, we are considering here, it is known (see e.g., [10, Theorem 4.1]) that

so that B-frees have asymptotic density ℳ B .

We write 〈 B 〉 for the multiplicative semigroup generated by B, that is the set of positive integers that can be written as a product of (possibly repeated) elements of B. By convention, 1 ∈ 〈 B 〉 , as 1 arises from the empty product. (For instance, if B = { p 2 : p ⁢ prime } , then 〈 B 〉 = { n 2 : n ∈ ℕ } .)

We introduce an arithmetic function μ B : ℕ → ℂ , analogous to the Möbius function μ, defined by

Observe that μ B and 𝟏 B ⁢ -free are multiplicative and relate via the multiplicative convolution

We denote by [ B ] ⊂ 〈 B 〉 the subset of 〈 B 〉 of elements n satisfying μ B 2 ⁢ ( n ) = 1 , i.e., those n ∈ 〈 B 〉 such that no b ∈ B divides n twice. (For instance, if B = { p 2 : p ⁢ prime } , then we have [ B ] = { n 2 : n ⁢ squarefree } .) We will often use without mention the fact that both [ B ] and 〈 B 〉 are closed under gcd and lcm , or equivalently, they are sublattices of the positive integers with respect to these two operations.

1.3 Fractional Brownian motion

We have mentioned that the squarefrees and more generally B-frees in a random interval ( n , n + H ] with n ≤ X are governed by a stationary ergodic process if H remains fixed. The number of B-frees in such an interval is ℳ B ⁢ H on average. This process has measure-theoretic entropy which becomes smaller the larger H is chosen to be, and thus does not seem very “random”. This may be compared to primes in short intervals ( n , n + ⌊ λ ⁢ log ⁡ X ⌋ ] , which contain on average λ primes. In such intervals, primes are conjectured to be distributed as a Poisson point process, and thus appear very “random”.

Nonetheless, a glance at Figure 1 comparing squarefrees to primes – along with consideration of the central limit theorems we have just discussed – reveals that, at a scale of H → ∞ , B-frees still retain some degree of randomness. It turns out that there is a natural framework to describe the “random” behavior of B-frees at this scale (analogous to the Poisson process for primes above), and this is fractional Brownian motion.

We give here a short introduction to fractional Brownian motion, as we believe this perspective sheds substantial light on the distribution of B-frees; however, the remainder of the paper is arranged so that a reader only interested in the central limit theorems of the previous sections can avoid this material.

Using Z ⁢ ( 0 ) = 0 , it is easy to see the covariance condition (1.3) is equivalent to

For a proof that such a stochastic process exists and is uniquely defined by this definition, see e.g. [36].

Classical Brownian motion is a fractional Brownian motion with a Hurst parameter of γ = 1 / 2 . If γ > 1 / 2 , increments of the process are positively correlated, with a rise likely to be followed by another rise, while if γ < 1 / 2 , increments of the process are negatively correlated.

Donsker’s theorem is a classical result in probability theory showing that a random walk with independent increments scales to Brownian motion (see [5, Section 8]). We prove an analogue of Donsker’s theorem for counts of B-frees using the following set up. We select a random starting point n ≤ X at uniform and define the random variables ξ 1 , ξ 2 , … in terms of n by

Set

where { τ } denotes the fractional part.

For integer τ, this is a random walk which increases on B-frees and decreases otherwise, and for non-integer τ, the function Q ⁢ ( τ ) linearly interpolates between values; thus Q ⁢ ( τ ) is continuous.

Figure 2

A graph of Q ⁢ ( τ ) and an analogue for the primes in the short interval ( n , n + H ] with n = 875624586 and H = 3000 . (a) illustrates a walk which increments by 1 - 6 / π 2 if u is squarefree, and - 6 / π 2 otherwise. (b) illustrates a walk which increments by 1 - 1 / log ⁡ u if u is prime, and - 1 / log ⁡ u if u is composite.

(a)

A random walk on squarefrees

(b)

A random walk on primes

Figure 3

A fractional Brownian motion and Brownian motion respectively, randomly generated using Mathematica, to be compared with the previous figure.

(a)

γ = 0.25 (fBm)

(b)

γ = 0.5 (Brownian motion)

Our proof of this result follows similar ideas as the proof of Theorem 1.10.

Note that α / 2 < 1 / 2 , so only a fractional Brownian motion with negatively correlated increments can be induced this way. It would be very interesting to understand functional limit theorems of this sort in the context of ergodic processes related to the B-frees described in e.g. [3, 8, 9, 10, 24, 27, 30]. There seem to exist only a few other constructions in the literature of a fractional Brownian motion as the limit of a discrete model, e.g. [1, 11, 18, 38, 44].

1.4 Notation and conventions

Throughout the rest of this paper, we allow the implicit constants in ≪ and O ⁢ ( ⋅ ) to depend on k when considering a k-th moment, and implicit constants are always for a fixed sieving set B. Later, we will introduce a weight function φ, and implicit constants depend on φ as well. Throughout the paper, where B has an index α, for simplicity, we will assume that α ∈ ( 0 , 1 ) . Some proofs would remain correct if α = 0 or 1, but the proofs of our central results would not be. We use the notation a ∼ x as a subscript in some sums to mean x < a ≤ 2 ⁢ x . In general, we follow standard conventions; in particular, e ⁢ ( x ) = e i ⁢ 2 ⁢ π ⁢ x , ∥ x ∥ denotes the distance of x ∈ ℝ to the nearest integer, ( a , b , c ) is the greatest comment divisor of a, b, and c, while [ a , b , c ] is the least common multiple.

1.5 The structure of the proof

There is a heuristic way to understand the Gaussian variation of N B ⁢ -free ⁢ ( n , H ) . Note that

The contribution of the first summand is close to the value ℳ B ⁢ H around which N B ⁢ -free ⁢ ( n , H ) oscillates. On the other hand, the functions n ↦ { ( n + H ) / d } - { n / d } are mean-zero functions of period d and thus are linear combinations of terms e ⁢ ( n ⁢ ℓ / d ) for 1 ≤ ℓ ≤ d - 1 . Upon reducing the fractions ℓ / d by the maximal divisor b ∣ ( ℓ , d ) with b ∈ 〈 B 〉 , we see that

is approximated by a linear combination of terms e ⁢ ( n ⁢ ℓ / d ) for ( ℓ , d ) B-free and d ∈ [ B ] .

Heuristically, if X is large and n ∈ [ 1 , X ] is chosen uniformly at random, one may expect the terms { e ⁢ ( n ⁢ ℓ / d ) : ( ℓ , d ) ⁢ B ⁢ -free } to behave like a collection of independent random variables, and this would imply the Gaussian oscillation of N B ⁢ -free ⁢ ( n , H ) .

Nonetheless, we do not quite have independence; instead, roughly speaking, one may use the same Fourier decomposition to relate the k-th moment M k , B ⁢ ( X , H ) to (weighted) counts of solutions to the equation

where ∥ ℓ i / d i ∥ < 1 / H for all i. Indeed, the realization that M k ⁢ ( X , H ) for the squarefrees are related to these counts appears already in [17].

It can be seen that Gaussian behavior will then follow from most solutions to the above equation being diagonal, meaning there is some pairing ℓ i / d i ≡ - ℓ j / d j ⁢ mod ⁢  1 for all i, and this is what we demonstrate. Our main tool is the Fundamental Lemma and its extensions (see Lemmas 2.6, 4.1, and 5.5), developed by Montgomery–Vaughan [34] and later used by Montgomery–Soundararajan [33] to prove a conditional central limit theorem for primes in short intervals.

However, this strategy if used by itself is not sufficient to prove a central limit theorem; the Fundamental Lemma was already known to Hall who used it to obtain his upper bound M k ⁢ ( X ; H ) ≪ H ( k - 1 ) / 2 . The reason this strategy does not work as it did for Montgomery–Soundararajan is that M k , B ⁢ ( X ; H ) has size roughly H α ⁢ k / 2 ; for primes, the k-th moment of a short interval count is (conditionally) much larger. Thus, in order to recover a main term, our error terms must be shown to be substantially smaller than for the primes.

We obtain an upper bound for the number of off-diagonal solutions to (1.5) by bringing in two ideas in addition to those used by Montgomery–Soundararajan. The first is due to Nunes, who showed in the recent paper [37] that solutions to (1.5) make a contribution to M k ⁢ ( X ; H ) only when d i is larger than H 1 - o ⁢ ( 1 ) for all i and the least common multiple of the d i is larger than H k / 2 - o ⁢ ( 1 ) ; this argument appears in Section 5.1. The second is an observation that bounding a term T 4 which appears in a variant of the Fundamental Lemma (Lemma 5.5) requires a more delicate treatment than that which appears in [34, Lemma 8]. However, it can be accomplished in our context by counting solutions to a certain congruence equation, which in turn can be estimated using an averaging argument relying crucially on the Pólya–Vinogradov inequality; this is done in Section 5.2. See Remark 5.10 for further discussion of how the more direct, but less quantitatively precise, approach of [34, Lemma 8] is insufficient in our context.

In order to prove that counts of B-frees scale to a fractional Brownian motion, it is not sufficient to consider the flat counts N B ⁢ -free ⁢ ( n , H ) , but instead, we must consider the weighted counts

where φ belongs to a class of functions that includes step-functions. A proof of a central limit theorem for flat counts remains essentially unchanged as long as φ is of bounded variation and compactly supported in [ 0 , ∞ ) . Convergence to a fractional Brownian motion as a corollary of this central limit theorem is discussed in Section 7.

2.1 The Fundamental Lemma and other preliminaries

We prove Proposition 2.1 in the next subsection, but first we must introduce a few tools.

For t ∈ ℝ and Y ≥ 1 , we let

Note that E H is Φ H ⁢ ( t ) for φ = 𝟏 ( 0 , 1 ] . One has E H ⁢ ( t ) ≪ F H ⁢ ( t ) for all t, where

and recall ∥ t ∥ is the distance from t to the nearest integer. The next lemma studies the first and second moment of F H . The second moment was studied in [34, Lemma 4], and the first moment is implicit in [37, Lemma 2.3]; we include a proof for completeness.

We introduce the arithmetic function

The next lemma is a variation on an estimate of Nunes [37, Lemma 2.4], who treated the corresponding result when B = { p m : p ⁢ prime } with m ≥ 2 .