Values of the Dickman-de Bruijn function $\rho(u)$

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Numbers

$u$

$\rho(u)$

comment: The boundary value. The function is identically $1$ on $0\leq u\leq1$.

0.3068528194400546905827678785418234319244998656397447458793199905066063780303052843941366730035813125

0.04860838829113156690718303934340742135432958047814054231680528505148823573593247200409129337116770797

0.004910925647760832352739150923615186032484297417692945977961657528030631494153505349840165957891479305

0.0003547247004560397298338945107706235609516436105726162701669036821477003206922126803906929382184528416

0.00001964969635395528965175498612920452289459671980962327267045628854204220362614428655654864830520116536

8.745669953293916695580283572769972173380471976458048319843283490701751507455205957369890872419066466e-7

3.232069304226103772599785361728216157619475162802409072273440324202075270546185324928313771781708734e-8

1.016248282737836546534853935695695783824439958658074404463809039602732886867963160647959275185158800e-9

10:

2.770171837725958988758121200634342326343006650115606506625469924608958391571159227442464413125831985e-11

3/2:

0.5945348918918356180219868845356508634280095765375058023859856758558993287510857487322475721826865988

5/2:

0.1303195618322507456114389443076067397200331776511915871698896030537865596281599029547514661323311788

7/2:

0.01622959324323599163094189604678439986680615911030965241451229416257007039186053275168936549155862219

Definition

The Dickman-de Bruijn function $\rho$ is the continuous function with $\rho(u)=1$ for $0\leq u\leq1$ and $u\rho'(u)=-\rho(u-1)$ for $u>1$ [4] [1]. This table gives $\rho(u)$ at rational smoothness ratios $u$.

Parameters

$u$

— smoothness ratio ($u\geq0$)

Formulas

(1)

For $1\leq u\leq2$, $\rho(u)=1-\log u$.

(2)

For fixed $u\geq1$, $\lim_{x\to\infty}\Psi(x,x^{1/u})/x=\rho(u)$, where $\Psi(x,y)$ counts the positive integers $n\leq x$ whose prime factors are at most $y$ [2].

Comments

(3)

The limit in (2) is the fixed-$u$ smooth-number theorem.

(4)

$\rho$ is identically $1$ on $0\leq u\leq1$, so the table begins at $u=1$ and omits that constant stretch.

(5)

The method of steps for the delay equation [3] 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]

N. G. de Bruijn, The asymptotic behaviour of a function occurring in the theory of primes, Journal of the Indian Mathematical Society, New Series 15 (1951), 25-32. https://research.tue.nl/en/publications/the-asymptotic-behaviour-of-a-function-occuring-in-the-theory-of-

[2]

Adolf Hildebrand and Gerald Tenenbaum, Integers without large prime factors, Journal de théorie des nombres de Bordeaux 5 (1993), no. 2, 411-484. (doi) (zbMATH) (MR)

[3]

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

[4]

Wikipedia: Dickman function

[5]

SageMath reference: Dickman rho function

Similar tables

Values of $\xi(u)$ in the Dickman-de Bruijn estimate — gives the saddle-point parameter in de Bruijn's asymptotic estimate for $\rho(u)$

Golomb-Dickman constant $\lambda$ — equals $\int_0^\infty \rho(u)/(1+u)^2\,du$

Values of the linear sieve function $F(s)$ — is another analytic factor in sieve estimates, indexed by a sifting ratio rather than by a smoothness ratio

Data properties

Entries are of type: real number

How well the digits are known: proven

Table is complete: no (it preserves the delay breakpoints $u=1,\ldots,10$ 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.