A link between error terms when counting smooth and rough numbers

Abstract: We establish a relationship between error terms appearing in estimates for the counting functions of smooth and rough numbers. We then apply this link to obtain an explicit upper bound for the error term in de Bruijn's approximation $Λ$ for the count of smooth numbers, from an explicit upper bound, due to Fan, for the error term in a variant of de Bruijn's estimate for the count of rough numbers.

• The paper establishes a rigorous link connecting error terms in the asymptotic estimates for y-smooth and y-rough numbers using integral transforms.
• It employs Möbius inversion and Laplace transform techniques to derive explicit error bounds, with results applicable under both classical assumptions and the Riemann Hypothesis.
• The analytical framework offers practical insights for sieve theory, cryptographic algorithms, and further research on friable integer statistics.

Error Terms in Counting Smooth and Rough Numbers: Analytical Connections and Explicit Bounds

Overview

The paper "A link between error terms when counting smooth and rough numbers" (2604.22058) develops a rigorous relationship connecting the error terms arising from classical and refined asymptotics for the counting functions of $y$-smooth and $y$-rough integers. These results establish a formal equivalence between the analytic behavior of error terms in estimates for $\Psi(x, y)$ (counting $y$-smooth numbers) and $\Phi(x, y)$ (counting $y$-rough numbers), yielding explicit bounds via Möbius inversion and Laplace transform techniques. The framework synthesizes classical results (de Bruijn’s and Buchstab’s asymptotics), prior explicit bounds, and new structural theorems, with strong implications for analytic number theory and sieve methods.

Definitions and Classical Estimates

Let $y \geq 2$, and $u = \frac{\log x}{\log y}$. The paper considers:

• $\Psi(x, y)$: the number of $y$-smooth numbers up to $y$0 (all prime factors $y$1),
• $y$2: the number of $y$3-rough numbers up to $y$4 (all prime factors $y$5),

with classical asymptotics: $y$6 where $y$7 is the Dickman function and $y$8 is the Buchstab function.

Refined approximations are used:

• De Bruijn's $y$9 for $\Psi(x, y)$0,
• $\Psi(x, y)$1 and $\Psi(x, y)$2 for $\Psi(x, y)$3.

Main Theorem: Analytical Relationship between Error Terms

Let $\Psi(x, y)$4 and $\Psi(x, y)$5. Define $\Psi(x, y)$6, where $\Psi(x, y)$7 denotes $\Psi(x, y)$8-smooth numbers. The central result establishes:

$\Psi(x, y)$9

$y$0

These identities demonstrate that the error terms for smooth and rough number counts are intertwined via integral transforms involving $y$1 and $y$2, facilitating the transfer of bounds and analytic structure between them.

Explicit Error Bounds and Corollaries

Via explicit estimates for $y$3 due to Fan, and advanced bounds for $y$4:

• Without assuming RH:

$y$5

• Assuming RH:

$y$6

These bounds are valid for large $y$7, with trivial bounds for small $y$8 given by $y$9.

Corollaries demonstrate that the explicit bounds on $\Phi(x, y)$0 (derived from $\Phi(x, y)$1) propagate through the analytic relationships to yield corresponding bounds on $\Phi(x, y)$2 and $\Phi(x, y)$3, and vice versa, with multiplicative loss only a factor of $\Phi(x, y)$4, the reciprocal of the prime product up to $\Phi(x, y)$5.

Further, the paper proves that for exponential-type bounds ($\Phi(x, y)$6), these transfer bidirectionally between $\Phi(x, y)$7 and $\Phi(x, y)$8 with equivalence. This is verified by Laplace transform analysis and Möbius inversion.

Sieve-Theoretic and Analytical Implications

The results establish deep analytic connections, implying that improved error estimates in one regime (smooth or rough counts) automatically constrain the error in the other. This provides powerful analytic tools for sieve-theory, uniform distribution, and probabilistic number theory. The methods leverage the structure of the Dickman and Buchstab functions, the saddle-point method, and Möbius inversion, underpinning much of modern analytic number theory.

The identities and bounds also yield sharper estimates for practical computations of smooth and rough numbers, relevant to cryptographic algorithms (factoring, discrete logarithms), and the design and analysis of numerical sieves.

Future Directions

The new analytic framework opens several avenues for further theoretical and computational improvement:

• Potential refinement of explicit bounds for error terms in both smooth and rough regime by leveraging deeper properties of $\Phi(x, y)$9 and $y$0.
• Optimization of sieve algorithms using the structural duality between error terms.
• Investigation of possible generalizations to other friable/rough-type number sequences and higher correlations.
• Tightening bounds under additional hypotheses (e.g., generalized RH), and examining implications for random models and prime distribution.

Conclusion

This paper rigorously establishes explicit analytic relationships and bounds between the error terms for counting smooth and rough numbers, providing a unified structural theory with practical and theoretical implications for analytic number theory and sieve methods. The results enable systematic cross-transfer of bounds, optimize error control, and suggest new directions for advancing the quantitative and computational analysis underlying friable integer statistics.