On the distribution of powered numbers

1 Introduction

Motivated by questions about diophantine equations and the a b c -conjecture, Mazur [1] proposed to smooth out the set of positive l th powers in a multiplicative way, by what he named powered numbers. To introduce the latter, let k ( m ) denote the largest square-free divisor of the natural number m . Mazur’s powered numbers (relative to the real number l > 0 ) are the numbers m ∈ N with k ( m ) l ≤ m . Note here that l need not be integral in this definition, but for l ∈ N , the powered numbers (relative to l ) contain the l th powers. It is proposed in [1] to replace, within a given diophantine equation, an l th power by the corresponding powered numbers and to consider the resulting equation between powered numbers as the associated “rounded” diophantine equation.

In this note, we analyse the distribution of powered numbers. We find it more appropriate to work with the real number ϑ = 1 ∕ l . The condition k ( m ) l ≤ m is then expressed equivalently as k ( m ) ≤ m ϑ , and for any ϑ > 0 , we define the set

Thus, Mazur’s powered numbers (relative to l ) are exactly the elements of A ( 1 ∕ l ) . For approaches to rounded diophantine problems, it is relevant to determine the density of the set A ( ϑ ) . We therefore consider the number S ϑ ( x ) of elements in A ( ϑ ) that do not exceed x . It is not difficult to see that for any 0 < ϑ < 1 , the number S ϑ ( x ) obeys the inequalities

whenever ε is a given positive real number and x is large in terms of ε . It now transpires that for ϑ = 1 ∕ l , the powered numbers are not much denser than the l th powers. A weaker version of (1.1) occurs in the study of Mazur [1] who refers to Granville, showing him an “easy” argument supposedly confirming the inequalities x ϑ − ε ≪ S ϑ ( x ) ≪ x ϑ + ε . In Mazur’s article, there is no indication how this would go but the simplest argument we know allows one to take ε = 0 in the lower bound. To substantiate this claim, fix a number ϑ ∈ ( 0 , 1 ) . Define l ∈ N by 1 ∕ l < ϑ ≤ 1 ∕ ( l − 1 ) , and t ∈ R by l − t = 1 ∕ ϑ . Then, t ∈ ( 0 , 1 ] and

Let W ( x ) be the number of natural numbers w that have a representation

with n , m square-free and constrained to the intervals

The numbers w counted by W ( x ) uniquely determine the square-free numbers n , m with (1.3) and (1.4). This follows from unique factorisation. Hence, W ( x ) equals the number of pairs n , m of square-free numbers satisfying (1.4). In particular, we see that W ( x ) ≫ x ϑ .

Next, we observe that (1.3) gives k ( w ) ≤ 2 n m , and therefore, k ( w ) ≤ 2 ( 8 − l x ) ϑ . By (1.2) and (1.4), we also have w ≥ 2 − ( 2 l − 1 ) x . It follows that k ( w ) ≤ w ϑ . Hence, numbers counted by W ( x ) are also counted by S ϑ ( x ) . This yields S ϑ ( x ) ≥ W ( x ) , and the left inequality in (1.1) follows at once.

For the upper bound in (1.1), one may refer to authorities like Tenenbaum [2, Theorem II.1.15]. However, the shortest argument available to us uses Rankin’s trick within the following chain of obvious inequalities:

In this argument, one needs to realise that the series ∑ k ( m ) − 1 m − ε converges, but this is immediately seen after converting this series to an Euler product.

For some of the finer questions on rounded diophantine equations, such as the asymptotic distribution of their solutions, the rough bounds displayed in (1.1) do not suffice. Instead, one desires an asymptotic formula for S ϑ ( x ) , and our primary goal in this note is to supply such a formula. Let 1 ≤ y ≤ x . Following Robert and Tenenbaum [3], we write N ( x , y ) for the number of m ∈ N with m ≤ x and k ( m ) ≤ y . The numbers counted here are their nuclear numbers. It seems natural to expect that the trivial upper bound S ϑ ( x ) ≤ N ( x , x ϑ ) should not be very wasteful. However, it was conjectured by Erdős^[1] (1962) and proved instantly by de Bruijn and van Lint [4] (1963) that

For a quantitative version of this estimate, see [3, Théorème 4.4]. These results suggest that the order of magnitude of S ϑ ( x ) is somewhat smaller than that of N ( x , x ϑ ) , and this is indeed the case.

Before we make this precise, we recall an estimate for N ( x , y ) . We are primarily interested in the case y = x ϑ with ϑ > 0 , but we work in the wider range y > exp ( ( log x ) 2 ∕ 3 ) . The multiplicative function

and the function F : [ 0 , ∞ ) → [ 0 , ∞ ) defined by

are featured in the uniform asymptotic formula

that is contained in [3, Proposition 10.1]. As we shall see momentarily, for each pair ϑ , x with 0 < ϑ < 1 and x ≥ 2 , there is exactly one real number α = α ϑ ( x ) > 0 with

We are now in a position to state our first result.

This result calls for several comments. First, we take y = x ϑ in (1.7) and substitute the resulting equation

within (1.10) to infer that

This is an analogue of (1.5), in quantitative form that is of strength comparable to [3, Théorème 4.4].

Our second comment concerns the implicitly defined function α ϑ ( x ) . It originates in the Dirichlet series

that converges absolutely in Re s > 0 and therefore has no zeros in this half plane. For real numbers σ > 0 , we have G ( σ ) > 0 . We may then define

Note that g extends to a holomorphic function on the right half plane, and one computes the logarithmic derivative of G ( s ) from the Euler product representation (1.12) to

Note that the sum on the right-hand side here coincides with the sum in (1.8). On differentiating again, it transpires that the real function g ′ : ( 0 , ∞ ) → R is increasing. Considering σ → 0 and σ → ∞ , one finds that its range is the open interval ( − ∞ , 0 ) . We conclude that for a given v > 0 , there is a unique positive number σ v with

From [3, Lemme 6.6], we infer that for v ≥ 3 , one has

Here, we choose v = ( 1 − ϑ ) log x and then have α ϑ ( x ) = σ v . In particular, we see that (1.9) and (1.13) imply (1.10). Thus, it only remains to prove (1.9).

Our last comment concerns the actual size of S ϑ ( x ) . This requires some more information on the function F . From [3, (2.12)], we have the asymptotic relation

By inserting (1.14) into (1.10), we deduce that there exists some positive number β ( x ; ϑ ) > 0 with

and the property that for any fixed ϑ ∈ ( 0 , 1 ) one has

Note that (1.15) yields another proof of (1.1).

We now turn to local estimates for S ϑ ( x ) . In our case, this amounts to comparing the respective behaviour of S ϑ ( z x ) and S ϑ ( x ) uniformly for large x , when z is in some sense sufficiently close to 1. Such estimates are often obtained with the saddle-point method, and we follow this route here, too. In a suitable range for z , the fraction S ϑ ( z x ) ∕ S ϑ ( x ) may be approximated by a simple function of z .

Finally, we consider the counting function for a variation of the powered numbers. For given ϑ ∈ ( 0 , 1 ) and Θ ∈ R , we consider

Note that S ϑ ( x ) = S ϑ , 0 ( x ) . The set of integers such that k ( n ) ≤ n ϑ ( log n ) Θ plays a prominent role in our companion article [5], and therefore, we provide an estimate for S ϑ , Θ ( x ) . It turns out that the conditions k ( n ) ≤ n ϑ and k ( n ) ≤ n ϑ ( log n ) Θ are relatively close and that the ratio S ϑ , Θ ( x ) ∕ S ϑ ( x ) is roughly of size ( log x ) Θ .

2 Proof of Theorem 1

In this section, we derive Theorem 1. Before we embark on the main argument, we fix some notations and recall a pivotal result concerned with the distribution of square-free numbers. This involves the function ψ ( m ) as defined in (1.6), the Möbius function μ ( m ) , and for a parameter 0 ≤ γ ≤ 1 2 at our disposal, the product

One then has the estimate (see [3, (10.1)])

that holds uniformly relative to the square-free number k and the real parameters z , γ in the ranges z ≥ 1 , 0 ≤ γ ≤ 1 2 .

The first steps of our argument follow the pattern laid out in [3, Sect. 10]. Unique factorisation shows that for all natural numbers n , there exists exactly one pair of coprime natural numbers l , m with μ ( l ) 2 = 1 and n = l m k ( m ) . Note that the two conditions ( l , m ) = 1 and μ ( l ) 2 = 1 are equivalent to the single condition μ ( l k ( m ) ) 2 = 1 . Furthermore, one has k ( n ) = l k ( m ) . With ϑ ∈ ( 0 , 1 ) now fixed, it follows that S ϑ ( x ) equals the number of ( l , m ) ∈ N 2 satisfying the conditions

These last three conditions we recast more compactly as

From now on, the number κ = ϑ ∕ ( 1 − ϑ ) features prominently, and we also put y = x ϑ . Note that

Hence, we consider the ranges m ≤ x ∕ y and x ∕ y < m ≤ x separately. By (2.2) and (2.3), this leads to the decomposition

in which

We apply (2.1) with k = k ( m ) to both inner sums and obtain

where

It turns out that R 1 and R 2 are small. In order to couch their estimation, as well as the analysis of other error terms that arise later, under the umbrella of a single treatment, we choose parameters γ and σ with 0 < γ < σ ≤ 1 2 and introduce the series

If one turns the sum over m into an Euler product, the factor at the prime p becomes 1 + O σ ( p γ − 1 − σ ) , so the conditions σ > γ > 0 ensure convergence. It is routine to show that

In fact, by Rankin’s trick,

For m ≤ x ∕ y , one has m κ ≤ y , and it follows that R 1 ≤ E .

Likewise, one confirms R 2 ≤ E by observing that 1 − γ > 1 2 ≥ σ so that

The appearance of k ( m ) in the summation conditions on the right-hand sides of (2.5) and (2.6) is a nuisance, and we proceed by removing these. If the condition k ( m ) ≤ m κ is removed from the sum in (2.5), one imports an error no larger than

Here, the penultimate inequality is obtained by the argument that completed the estimation of R 1 . Similarly, if the condition m k ( m ) ≤ x is removed from the summation condition in (2.6), then the resulting error does not exceed

say. By Rankin’s trick and (2.8),

Collecting together, we deduce from (2.5) and (2.6) the asymptotic relations

and by (2.4), we infer that

Note that the sum on the right is a partial sum of a convergent series. If one completes the sum, then it is immediate that the error thus imported is bounded by R and hence by E . We have now reached the provisional expansion

It remains to estimate E . In its definition (2.7), we encounter a sum over a multiplicative function, and so

Now, since p γ − 1 2 ≤ 1 , γ < σ , p σ − 1 ≥ σ log p , and since the sum ∑ p 1 p log p converges

As in the discussion following (1.13), we write v = ( 1 − ϑ ) log x = log x y , and then choose σ = σ v and γ = σ v − 1 log y . Then γ < σ , and for large x , we have σ < 1 2 . By (1.12), one has

Now, on recalling (1.13),

while one also has

On collecting together, this shows that

From [3, (2.11)], we deduce that

and hence,

With the choice of y and v , one has σ v = α ϑ ( x ) . Moreover, log y and v have the order of magnitude log x so that the last inequality now reads

Our final task is to compare our estimate for E with the size of the sum on the right of (2.9).

Recall that in view of (1.1) and (1.11), S ϑ ( x ) and N ( x , x ϑ ) are of comparable size. In order to mimick the estimate (1.11), we introduce the function

so that (2.9) now reads

Our aim is to give an estimate of H ϑ ( x ) by using the saddle-point method, and to describe more precisely the behaviour of the function H ϑ ( x ) ∕ F ( ( 1 − ϑ ) log x ) as x → ∞ .

We may now complete the proof of Theorem 1. Using the lemma and (1.13), we obtain

so that the estimate (2.10) implies