Approximation of smooth numbers for harmonic samples: a Stein method approach

Although the problem of obtaining sharp estimates for $\Psi(n,m)$ will not be addressed in this paper, we would like to bring the reader’s attention to some ideas that might be useful for the treatment of this particular problem. First we notice that a good approximation for $\mathbb{P}[\psi(H_{n})\leq m]$ is a fundamental first step for this task. Indeed, in virtue of (1.3), sharp estimations of $\Psi(n,m)$ can be obtained from their counterpart for $\Psi_{H}(n,m)$. In order to verify this, we write

Splitting the event $\{\psi(H_{n})\leq m\}$ into its components induced by the partition

and using the fact that for all $h\in\mathbb{N}$, it holds that

where $\pi:\mathbb{R}_{+}\rightarrow\mathbb{R}$ denotes the prime-counting function

we deduce the inequality

By a conditioning argument, the above identity yields

From the above discussion, we conclude that a sharp approximation for the distribution function of $\psi(H_{n})$ (or equivalently, for the function $\Psi_{H}$) can be used to transfer approximations of

to approximations for the distribution of $\psi(H_{n}Q_{n})$, and ultimately, to $\Psi(n,m)$. Arguably, the most natural approach for carrying out an analysis for (1.4) would be to utilize the available estimations regarding the prime counting function. In particular, by [23], the inequality

holds for $n\geq 229$ (see also [4, Section 3]), suggesting that an estimation of (1.4) could be obtained through an analysis of

The above term can be approximated, contingent on sharp estimates for

for

Approximations on this object are closely related to the description of the limiting behavior of the measure $\mu_{n,m}$, obtained as the distribution of $n/H_{n}$, conditional to $\psi(H_{n})\leq m$. We conjecture that the measure $\mu_{n,m}$ concentrates around its mean and, up to a suitable normalization constant, admits an asymptotic limiting distribution. We intend to address this topic in a forthcoming paper.

Now we turn our attention to the function $\Psi_{H}$, which is the main focus of our investigation. Similarly to $\Psi$, the description of the asymptotic behavior of $\Psi_{H}$ turns out to be closely related to the so-called Dickman function $\rho$, defined as the unique solution to the initial value problem of the delay type

However, the structure of dependence on $m$ and $n$ through $\rho$ for the approximation of $\Psi_{H}(m,n)$, is different from its counterpart for $\Psi(m,n)$. In particular, our main result, presented in Theorem 3.1, combined with Theorem 2.2, imply that the quotient

converges to one as the quotient $\log(n)/\log(m)$ tends to infinity. In particular, $\Psi(m,n)$ is not asymptotically equivalent to $\Psi_{H}(m,n)$ as $\log(n)/\log(m)$ tends to infinity. A second remarkable feature of $\Psi_{H}$ is the fact that it can be sharply estimated uniformly over $m,n\in\mathbb{N}$, unlike $\Psi(m,n)$, whose approximation has shown to be tractable only for regimes of $m$ and $n$ for which $\log(n)/\log(m)\leq f(n)$, with $f(x)$ being an adequate function satisfying $f(x)\leq\sqrt{x}$.

In the seminal paper [7], Dickman presented a groundbreaking result regarding the behavior of the function $\Psi$ and its relation to the solution to the delay equation (1.7). Dickman showed that for every $x\in\mathbb{R}_{+},$ the following convergence holds

This relation was further refined by Ramaswami in [18], who proved that for any fixed $x\geq 1$, there exists $C_{x}>0$, such that

These two results where subsequently explored thoroughly by many authors, seeking for extensions that allow for uniformity over $x$. To pin down precisely the aforementioned notion of uniformity, we observe that if we set the smoothness threshold $n^{1/x}$ appearing in (2.1), to be equal to $m$, we have that $x=\Upsilon(n,m)$, where $\Upsilon:\mathbb{N}\rightarrow\mathbb{R}$ is given by

so that the Dickman approximation (2.1), reads $\Psi\left(n,m\right)/n\approx\rho\circ\Upsilon(n,m)$, provided that we take $m=n^{1/x}$. Ramaswani’s approximation, on the other hand, becomes

One of the most significant generalizations to the above results was done by de Bruijn in [5], who proved Theorem 2.1 below

The mathematical community has put many efforts in finding sharp upper bounds asymptotically equivalent to (2.2), uniformly over a larger set than $\mathcal{S}_{\varepsilon}$. Next, we mention some of the most relevant ones for the purposes of our paper. We begin with the work by Hensley in [11], where it was shown that the relaxation

holds uniformly over considerably larger set than $\mathcal{S}_{\varepsilon}$, thus reducing the problem of estimating the exact value of $\Psi(n,m)$, to simply finding upper bounds for it. A crucial improvement to the de Bruijn estimation was done by Hildebrand in [13], where the following result was proved

Shortly after, Hildebrand presents in [12], an equivalence between the resolution of this problem and the Riemann hypothesis. The precise statement reads as follows

Although we do not treat directly with the function $\Psi$, the above results will serve as benchmarks for setting up appropriate parallelisms for $\Psi_{H}$.

In the sequel, $\gamma$ will denote the Euler-Mascheroni constant. Useful information on $\rho$ can be obtained from a probabilistic perspective by regarding the mapping $x\mapsto e^{-\gamma}\rho(x)$ as the density of a random variable. The verification of the condition $\int_{\mathbb{R}_{+}}e^{-\gamma}\rho(x)=1$ can be obtained from an adequate analysis of the Laplace transform of $\rho$, as explained in [14, Lemma 2.6.] (see also [15]). We say that a random variable $D$ is distributed according to the Dickman law if its cumulative distribution function is given by $e^{-\gamma}\mathcal{I}[\rho]$, where $\mathcal{I}[\rho]$ denotes the integral of $\rho$ over $[0,x]$. The main use we will give to this interpretation is the well-known characterization of the Dickman distribution in terms of its Bias transform (see [1]), which states that for every bounded and smooth function $f:\mathbb{R}_{+}\rightarrow\mathbb{R}$, it holds that

This identity can be easily obtained from the fact that if $D$ is distributed according to $e^{-\gamma}\rho$ and $U$ is a uniform random variable independent of $D$, then

The reader is referred to [17] for a proof of (4.2). The study of distributional approximations of random variables $X$ by means of sharp estimations on $\mathbb{E}[Xf(X)]$, for $f$ ranging over an adequate collection of test functions initiated with the work by Stein in [20] in the study of Gaussian approximations for sums of weakly dependent random variables and has been developed for many different distributions, including exponential, beta, gamma, Poisson, geometric and Dickman, among many others (see [16, 3, 9, 8]). A very useful adaptation of this methodology to the realm of weighted sums of random variables was presented in Theorem 1.4 and Theorem 1.5 in [2], whose proof serves as starting point for our application in hand. Replacing the role of $f$ in (4.1) by $f^{\prime}$, one obtains the following characterization, which can be found in [1, Lemma 3.2]

Let $\mathcal{C}$ denote the collection of test functions

Taking into consideration the above lemma, we implement Stein’s heuristics in the context for the Dickman law, which roughly speaking, aims to conclude an approximation of the type $\mathbb{E}[h(W)]\approx\mathbb{E}[h(D)]$ for $h\in\mathcal{C}$ by an approximation of the type

for $f$ belonging to a family of test functions determined by $\mathcal{C}$. This idea can be formalized by considering the Stein equation

where $h_{z}(x):=\mathbbm{1}_{[0,z]}(x)$. Evaluation at $x$ equal to the variable of interest, taking expectation and then supremum with respect to $z$, then yields a tractable expression for the distance in Kolmogorov of the variable of interest and the Dickman law. The solution to (4.3), along with a description of its regularity properties is presented in Theorem 1.9 and Lemma 3.3, from [2]. These results are summarized in Lemma 4.2 below.

Next we present some elementary number theory estimations related to partial products and partial sums over the primes for the functions $(1-1/p)$ and $\log(p)/p$, respectively. Such results are attributed to Franz Mertens and the estimations are referred to as “first, second and third Mertens’ formulas”. The first one establishes that

The second Merten’s formula reads

where $c_{1}$ is a universal constant, while the third guarantees the existence of a constant $C>0$, such that

The reader is referred to [22], pages 15-17, for a proof of these estimations.

Let $\{\xi_{p}\}_{p\in\mathcal{P}}$ be a sequence of independent random variables with distribution given as in (3.3). For $k\in\mathbb{N},p\in\mathcal{P}$, let $\alpha_{p}(k)$ denote the largest $m$ satisfying $p^{m}|k$. Therefore $k$ is uniquely determined by $\alpha_{p}$’s through the prime factorisation $k=\prod_{p\in\mathcal{P}}p^{\alpha_{p}(k)}$. Recall the definitions of $L_{n}$ and $I_{n}$, given by (3.1). The following result, first presented in [4], describes the join distribution of the $\alpha_{p}(H_{n})$’s.