A link between error terms when counting smooth and rough numbers
Pith reviewed 2026-05-08 14:01 UTC · model grok-4.3
The pith
A link between error terms for smooth and rough number counts transfers an explicit bound from one to the other.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
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.
What carries the argument
The explicit relationship between the error terms in the counting functions for smooth numbers and rough numbers, which permits direct transfer of bounds.
If this is right
- An explicit upper bound is now available for the error term in de Bruijn's approximation Λ of smooth number counts.
- The bound is obtained directly from Fan's explicit upper bound on the corresponding error for rough numbers.
- The transferred bound holds under the analytic conditions already present in the earlier work.
- The same link can be used in either direction to move explicit bounds between smooth and rough counting problems.
Where Pith is reading between the lines
- The link may extend to other counting functions defined by constraints on prime factors, allowing similar bound transfers.
- Numerical checks of the bound for moderate-to-large x and y would give practical evidence of its tightness.
- This transfer technique could reduce the work needed to derive explicit errors for related problems in analytic number theory.
Load-bearing premise
The analytic conditions and prior estimates of de Bruijn and Fan are compatible enough that the error terms for the two counting problems can be related without introducing new uncontrolled discrepancies.
What would settle it
For sufficiently large x and y, if the observed error in de Bruijn's approximation for the number of smooth numbers up to x exceeds the transferred upper bound by a positive constant factor, the claimed link would fail.
read the original 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 $\Lambda$ 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.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The manuscript establishes a relationship between the error terms appearing in estimates for the counting functions of smooth numbers (via de Bruijn's approximation Λ) and rough numbers (via a variant of de Bruijn's estimate). It then applies this link to transfer an explicit upper bound for the rough-numbers error term, due to Fan, into an explicit upper bound for the error term in the smooth-numbers case.
Significance. If the claimed relationship holds, the work supplies a direct mechanism for transferring explicit bounds between these two counting problems in analytic number theory. A notable strength is the parameter-free derivation of the link, which operates under the same analytic conditions (ranges of x and y, zero-free regions) already present in the source papers of de Bruijn and Fan, without introducing new hypotheses or data-dependent fitting.
minor comments (3)
- [§1] §1 (Introduction): the ranges of x and y for which the relationship and the resulting bound are valid should be stated explicitly at the outset, matching the conditions under which Fan's bound is known to hold.
- [§2] §2 (statement of the link): the notation distinguishing the error term for smooth numbers from the variant error term for rough numbers is introduced without a side-by-side comparison table; adding one would clarify the transfer step.
- [§3] §3 (application): the final explicit bound is stated but not compared numerically, even for a single small value of y, with any previously published explicit bounds for the same quantity.
Simulated Author's Rebuttal
We thank the referee for the positive assessment of our work, including the recognition of the parameter-free link between the error terms and the potential for transferring explicit bounds. The recommendation is for minor revision, but the report lists no specific major comments or requested changes.
Circularity Check
No significant circularity; derivation transfers external bound via independent link
full rationale
The paper derives a relationship between error terms in smooth-number and rough-number counting functions under the analytic hypotheses (ranges for x,y and zero-free regions) already present in de Bruijn's and Fan's prior estimates. It then substitutes Fan's explicit upper bound (an independent external result) into the new link to bound the error in de Bruijn's Λ. No step reduces by construction to a fitted parameter, self-definition, or self-citation chain; the central claim consists of the explicit transfer made possible by the derived link, which remains falsifiable against the cited estimates without tautology. This is the normal non-circular case of relating two externally benchmarked approximations.
Axiom & Free-Parameter Ledger
axioms (1)
- domain assumption Standard analytic estimates and error-term relations for smooth and rough number counting functions hold as established by de Bruijn and Fan.
Reference graph
Works this paper leans on
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[1]
de Bruijn, On the number of uncancelled elements in the sieve of Eratosthenes, Ned
N.G. de Bruijn, On the number of uncancelled elements in the sieve of Eratosthenes, Ned. Akad. Wet. Proc.53(1950) 803–812
work page 1950
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[2]
N. G. de Bruijn, On the number of positive integers≤xand free of prime factors > y,Nederl. Akad. Wetensch. Proc. Ser. A54(1951), 50–60
work page 1951
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[3]
Fan, Numerically explicit estimates for the distribution of rough numbers,J
K. Fan, Numerically explicit estimates for the distribution of rough numbers,J. Number Theory,260(2024) 120–150
work page 2024
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[4]
J. B. Rosser and L. Schoenfeld, Approximate formulas for some functions of prime numbers,Illinois J. Math.6(1962) 64–94
work page 1962
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[5]
Saias, Sur le nombre des entiers sans grand facteur premier,J
E. Saias, Sur le nombre des entiers sans grand facteur premier,J. Number Theory32 (1989), no. 1, 78–99
work page 1989
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[6]
G. Tenenbaum, Introduction to Analytic and Probabilistic Number Theory, Third Edition, Graduate Studies in Mathematics, Vol. 163, American Mathematical Society, 2015. Department of Mathematics, Southern Utah University, 351 West Univer- sity Boulevard, Cedar City, Utah 84720, USA Email address:weingartner@suu.edu
work page 2015
discussion (0)
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