Friable Numbers: Theory & Applications

• Friable numbers are integers whose largest prime factor is bounded by y, analyzed using asymptotic formulas and probabilistic laws.
• Saddle-point analysis, Laplace transforms, and delay-differential techniques provide precise estimates for functions like Ψ(x, y) and related sums.
• Applications range from improved Diophantine approximations and cryptographic designs to Gaussian limit laws in additive function distributions.

A positive integer $n$ is called $y$-friable (or $y$-smooth) if its largest prime factor $P^+(n)$ satisfies $P^+(n)\leq y$ (with the convention $P^+(1)=1$). The study of friable numbers lies at the intersection of analytic number theory, sieve theory, and probabilistic combinatorics, with applications ranging from arithmetic function averages and additive combinatorics to cryptography and Diophantine approximation. The distribution of friable numbers, their extremal statistics, and their behavior in various arithmetic settings are governed by a rich array of asymptotic formulas, probabilistic limit laws, and advanced analytic techniques.

1. Definitions, Notation, and Basic Counting Functions

Let $S(x, y) := \{ n \leq x : P^+(n)\leq y \}$ denote the set of $y$-friable integers up to $x$, and let $\Psi(x, y) = |S(x, y)|$ be its cardinality. For an arithmetical function $y$0, define the $y$1-friable sum as

$y$2

The parameter $y$3 (“degree of friability”) is fundamental and controls the typical “thickness” of the friable set.

The classical Dickman–de Bruijn function $y$4 governs the density of friable integers: $y$5 satisfying the differential–difference equation $y$6 for $y$7 (Moree, 2012).

The main asymptotic for friable integers established by de Bruijn is

$y$8

uniformly for $y$9, with far-reaching uniform refinements and error terms provided in subsequent work (Moree, 2012, Lichtman et al., 2017).

2. Key Asymptotics and Methods

The fundamental estimation of $y$0 employs complex-analytic, saddle-point, and Laplace-transform techniques (Moree, 2012, Lichtman et al., 2017). The saddle-point $y$1 is defined as the unique solution to

$y$2

and allows an explicit representation

$y$3

where $y$4 and $y$5 (Drappeau, 2013, Lichtman et al., 2017). The error analysis can be made fully effective, yielding explicit numerical intervals for $y$6 in large ranges of $y$7 (Lichtman et al., 2017).

For complex-valued arithmetical functions $y$8 whose Dirichlet series $y$9 for some $P^+(n)$0 and holomorphic $P^+(n)$1, asymptotic expansions of the form

$P^+(n)$2

hold, where $P^+(n)$3 solves a fractional delay-differential equation generalizing Dickman’s function (Bretèche et al., 2023, Bretèche et al., 2022).

3. Probabilistic and Statistical Properties

Friable integers exhibit Gaussian statistics for additive functions and divisor distribution. For $P^+(n)$4, where $P^+(n)$5 is a random divisor of $P^+(n)$6, the distribution is asymptotically normal: $P^+(n)$7 for almost all $P^+(n)$8, where $P^+(n)$9 and $P^+(n)\leq y$0 (a normalized fourth cumulant index) are as in (Drappeau et al., 2016). Averaged over $P^+(n)\leq y$1, the normalized divisor logarithms admit a central-limit theorem (Drappeau, 2015).

The distribution of the additive function $P^+(n)\leq y$2 (number of prime factors) among $P^+(n)\leq y$3 satisfies an Erdős–Kac law: for $P^+(n)\leq y$4,

$P^+(n)\leq y$5

uniformly in $P^+(n)\leq y$6 (Bretèche et al., 2023). Variance and expectation of $P^+(n)\leq y$7 conditioned on friability approach $P^+(n)\leq y$8 with explicit error terms.

Weighted random models on $P^+(n)\leq y$9, assigning probability proportional to $P^+(1)=1$0, lead to Gaussian bias phenomena: the limiting distribution of the normalized logs of random $P^+(1)=1$1-friable $P^+(1)=1$2 becomes standard normal, with explicit scale and shift (Tenenbaum, 2022).

4. Distribution in Arithmetic Progressions and Linear Forms

Friable numbers are, in specific regimes, well-distributed in arithmetic progressions. Uniform average bounds for

$P^+(1)=1$3

hold for $P^+(1)=1$4 and fixed $P^+(1)=1$5 (Drappeau, 2013). This extends Bombieri–Fouvry–Iwaniec-type exponents of distribution to sparse friable sets, and yields applications such as asymptotics for sums like $P^+(1)=1$6.

Recent advances demonstrate that for systems of $P^+(1)=1$7 affine-linear forms $P^+(1)=1$8, the count of $P^+(1)=1$9 in a convex body $S(x, y) := \{ n \leq x : P^+(n)\leq y \}$0 such that each $S(x, y) := \{ n \leq x : P^+(n)\leq y \}$1 is $S(x, y) := \{ n \leq x : P^+(n)\leq y \}$2-friable enjoys a full asymptotic formula,

$S(x, y) := \{ n \leq x : P^+(n)\leq y \}$3

proving the predicted “independence” heuristic for multilinear friable problems (Lachand, 2016).

The distribution of friable numbers in short intervals, arithmetic progressions, and more general structures (e.g., values of polynomials and shifted sets) has been further quantified with effective lower-bound sieves, enabling effective estimations even with a sparse level of distribution (Mounier, 2024).

5. Additive, Multiplicative, and Combinatorial Properties

Friable numbers resist certain additive and multiplicative decompositions. Results show that, for slowly growing $S(x, y) := \{ n \leq x : P^+(n)\leq y \}$4 (e.g. $S(x, y) := \{ n \leq x : P^+(n)\leq y \}$5), there are no infinite sets $S(x, y) := \{ n \leq x : P^+(n)\leq y \}$6 with $S(x, y) := \{ n \leq x : P^+(n)\leq y \}$7 such that $S(x, y) := \{ n \leq x : P^+(n)\leq y \}$8 covers all sufficiently large $S(x, y) := \{ n \leq x : P^+(n)\leq y \}$9-friable numbers, nor is there a multiplicative analog (Győry et al., 2020). Finiteness results for $y$0-unit equations enable these irreducibility theorems.

Explicit exponential sum estimates over friable numbers,

$y$1

are nontrivial for $y$2, enabling asymptotics for $y$3-friable solutions to $y$4 (Drappeau, 2013). In particular, uniform estimates such as

$y$5

are valid in the same regime.

Specialized mean value theorems for the Erdős–Hooley Delta function

$y$6

provide two-sided uniform bounds in the Hildebrand–Tenenbaum region $y$7, and precise asymptotics in “very friable” ranges (Martin et al., 2023).

6. Applications and Further Directions

Friable numbers are crucial in a variety of analytic number theory problems and applications:

• Waring’s problem and minor arc analysis: Precise estimates for Weyl sums over $y$8-friable $y$9 up to $x$0 in regimes $x$1 lead to asymptotic counts for representations of integers as sums of $x$2th powers of friable numbers, with the number of summands essentially matching the unrestricted case (Drappeau et al., 2016).
• Diophantine approximation: The set of $x$3-friable numbers achieves fine inhomogeneous Diophantine approximation exponents; e.g., for any irrational $x$4 and $x$5 below $x$6 or $x$7 (depending on the smoothness), there are infinitely many $x$8 with $x$9 and $\Psi(x, y) = |S(x, y)|$0 (Nath et al., 18 Mar 2026).
• Arithmetic function averages: The behavior of multiplicative and complex-valued functions over $\Psi(x, y) = |S(x, y)|$1 is governed by delay-differential systems, extending Selberg–Delange expansions to the friable regime (Bretèche et al., 2023, Bretèche et al., 2022).
• Sieve theory and “friable sieve”: Effective minorant sieve inequalities enable lower-bound results for counts of friable values of polynomials, binary forms, and shifted/friable-twin problems, at the edge of known levels of distribution (Mounier, 2024).
• Divisor distribution and Gaussian limit laws: The distribution of divisors and additive functions on $\Psi(x, y) = |S(x, y)|$2 exhibits optimal Gaussian behavior with explicit error rates and exceptional set measures (Drappeau et al., 2016, Drappeau, 2015).

Ongoing research further elucidates phase transitions at thresholds such as $\Psi(x, y) = |S(x, y)|$3 for the validity of the main Dickman–de Bruijn asymptotic (Bretèche et al., 2022), the limitations of the method in ultra-sparse regimes, and effective computation of $\Psi(x, y) = |S(x, y)|$4 in explicit numerical applications (Lichtman et al., 2017).

7. Tables: Principal Asymptotics and Distributions

Quantity	Asymptotic/Main Term	Validity Domain
$\Psi(x, y) = |S(x, y)|$5	$\Psi(x, y) = |S(x, y)|$6	$\Psi(x, y) = |S(x, y)|$7, $\Psi(x, y) = |S(x, y)|$8 (Moree, 2012)
$\Psi(x, y) = |S(x, y)|$9	$y$00	$y$01 (Drappeau, 2013)
Erdős–Kac law (friable)	$y$02	$y$03 fixed, $y$04 (Bretèche et al., 2023)
Divisor log CLT	$y$05	$y$06 large, $y$07 except for $y$08 (Drappeau et al., 2016)

The constants and ranges in these formulas are governed by the underlying analytic techniques—saddle-point analysis, delay-differential systems, and sieve-theoretic bounds.

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Friable numbers remain a central object of research, providing a test-bed for major analytic, probabilistic, and combinatorial techniques in number theory, with a suite of results now reaching full uniformity and precision across broad parameter regimes.