Values of the Buchstab function $\omega(u)$

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Numbers

$u$

$\omega(u)$

1/2

0.5643823935199817698057440404860588560251667114534184180402266698311312073232315718686211089988062292

0.5614582414068377374244183782971147076623276740309098614465484880816192055432876014378307007496579714

0.5614544682684567287363246889425285326620983483162800302968678707644020228811670126648449201606215930

0.5614596848016890453857407290415545292990093348345936324170409534788144194274186385747928001461618182

0.5614594806894847631436195446529916782578631714436753446494409067676190754966361811635894294979436975

0.5614594835344085231709984685685153333556755808799845705042446865033247303354093400695417269392091758

3/2:

2/3

5/2:

0.5621860432432657527912052461857396546287961693849976790456057296576402684995657005071009711269253605

7/2:

0.5608288644515888217081409622410368236057816049907761493582358191472449403154531431437978766645294358

Definition

The Buchstab function $\omega$ is the continuous function on $[1,\infty)$ with $\omega(u)=1/u$ for $1\leq u\leq2$ and $(u\omega(u))^\prime=\omega(u-1)$ for $u>2$ [2] [3]. This table gives $\omega(u)$ at roughness ratios $u$.

Parameters

$u$

— roughness ratio $\log x/\log y$ ($u\geq1$)

Formulas

(1)

$\omega(u)=1/u$ for $1\leq u\leq2$ [2].

(2)

$(u\omega(u))^\prime=\omega(u-1)$ for $u>2$ [2].

(3)

$u\omega(u)=1+\int_1^{u-1}\omega(t)\,dt$ for $u\geq2$ [3].

(4)

$\omega(u)=(1+\log(u-1))/u$ for $2\leq u\leq3$ (3).

(5)

$\lim_{u\to\infty}\omega(u)=e^{-\gamma}$ [2].

Comments

(6)

In rough-number estimates, $u=\log x/\log y$ measures the gap between the counting range $x$ and the excluded prime-factor bound $y$.

(7)

The function tends to $e^{-\gamma}$ as $u\to\infty$ (5), where $\gamma$ is the Euler-Mascheroni constant.

(8)

The method of steps for the delay equation [1] makes unit intervals the natural analytic pieces. One midpoint from each of the first three intervals supplies interior reference values without extending a dense rational mesh or merely adding more joins. The associated power thresholds are $y=x^{2/3},x^{2/5},x^{2/7}$. This bounded interior selection is exploratory; the reference supports the equation and interval structure, not separate demand for each midpoint.

Programs

(P1)

Python

from generate import generator
from numberdb._write import to_text

for params in generator.enumerate():
    print(params, to_text(generator.value(params, 100)["number"], 100))

References

[1]

George Marsaglia, Arif Zaman and John C. W. Marsaglia, Numerical solution of some classical differential-difference equations, Mathematics of Computation 53 (1989), 191-201. (doi)

Links

[2]

Wikipedia: Buchstab function

[3]

MathWorld: Buchstab Function

[4]

Tao, Chen's theorem notes: the linear sieve

[5]

Wikipedia: Sieve theory

Similar tables

Values of the linear sieve function $F(s)$ — another delay-differential analytic factor in linear-sieve estimates

Values of $\xi(u)$ in the Dickman-de Bruijn estimate — gives the saddle-point parameter for estimates involving the smooth-number counterpart to rough-number counts

Golomb-Dickman constant $\lambda$ — another constant from the Dickman-de Bruijn function and smooth-number asymptotics

Hardy-Littlewood singular series of prime tuples — stores arithmetic singular-series factors that occur beside analytic sieve factors

Twin prime constant $C_2$ — a prime-pair singular-series factor from the same sieve-theoretic neighbourhood

Bateman-Horn constants of monic quadratic polynomials — stores arithmetic constants for prime-value conjectures proved or approached by sieve methods

Named rational Euler products over primes — stores prime-product constants that often supply the arithmetic factor beside sieve functions

Data properties

Entries are of type: real number

How well the digits are known: proven

Table is complete: no (it preserves the domain endpoint and delay breakpoints $u=1,\ldots,8$ and adds $u=3/2,5/2,7/2$ as exploratory interior references in the first three unit intervals after the endpoint $1$)

How they were obtained:

All retained numeric strings, exact values and row annotations are unchanged. The standalone generate.py, balanced_values.py and selection.json reproduce the entire current selection. Each finite Arb enclosure at 733 bits must fit inside the interval claimed by the written last place; a second run uses 912 bits. New nonexact entries have 100 significant digits.

Every new entry has an independent mpmath quadrature, closed-form or inverse-function check at 150 and 200 decimal digits, with a nonzero known-integral control. Arguments are exact rationals or propagated balls from exact expressions; root arguments use strict sign-certified rational brackets. Lambert and saddle roots also pass endpoint sign tests on the written last-place interval. The delay method uses degree 320 and a separate uniform remainder propagated through integration; the second run raises the degree to 420. Earlier deep delay entries retain their historical independent validation; this run rechecks their enclosures and order agreement without claiming a new independent deep-delay computation. The measurements are in coverage-checks.json. Older attachments are retained for provenance; their previous enumerations are historical, and the active enumeration is selection.json.