Values of $\xi(u)$ in the Dickman-de Bruijn estimate

edit · history · discussion · files · short url · special functions number theory

Numbers

$u$

$\xi(u)$

comment: The positive branch used for $u>1$ has limiting value $0$ at $u=1$.

1.256431208626169676982737616609216326916416831701323711125894727048300478541051903533664750947250849

equals: $\lambda_{3,2}$
comment: This is the $r=2$ core-threshold minimiser $\lambda_{3,2}$.

1.903813694440383484710140360828135127280364804561987445879389818281945875445251742935897221209070264

equals: $\lambda_{4,2}$
comment: This is the $r=2$ core-threshold minimiser $\lambda_{4,2}$.

2.336662982263053881182850266714011154679504739557109039685225120116767568104385781543593285556392922

equals: $\lambda_{5,2}$
comment: This is the $r=2$ core-threshold minimiser $\lambda_{5,2}$.

2.660399058463684990432673112693838761316192793557399779133621108141380344526679670369701126317582048

equals: $\lambda_{6,2}$
comment: This is the $r=2$ core-threshold minimiser $\lambda_{6,2}$.

2.918300475783052591304221719797499459970061873861584609944495208244877889764383735414949103879265396

equals: $\lambda_{7,2}$
comment: This is the $r=2$ core-threshold minimiser $\lambda_{7,2}$.

3.132265450540420106833422460869303332477500804253648991520139627783433414358227583586491785309904738

equals: $\lambda_{8,2}$
comment: This is the $r=2$ core-threshold minimiser $\lambda_{8,2}$.

3.314877361786054930901651570191749329550314788552505601434533572470055998782613615635980490671720637

3.474019476966354505452070987403288659224133427022420159376911107356917224200938678254396618806431641

10:

3.614950427087530629681888723572661435176889879499709780542263575885132076037722882450886931283670500

11:

3.741349652356515880952925500625653961859467356437047879191127422890009883669962272286344364902196782

12:

3.855890308448959712952366737654264389572596946850658243654505887045967065647643973726513680199805710

3/2:

0.7626885608503389820439021764368102222879709347529379724068165212571283274895766670730823743109837147

5/2:

1.618788125264683184559765827258905623048980981186977513366824128381904573130926940976004047610963874

7/2:

2.138109224898635763921094054319228832184279393623334212884171400615023038106414735586297204460649494

9/2:

2.508726441467471078712734347455562125731353107775127114612782791600653683069932287325654123689115897

Definition

Let $\xi(1)=0$, and for $u>1$ let $\xi(u)$ be the positive root of $e^{\xi}-1=u\xi$. This is the saddle-point parameter in de Bruijn's asymptotic estimate for the Dickman-de Bruijn function $\rho(u)$ [1] [2] [3].

Parameters

$u$

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

Formulas

(1)

For $u>1$, $\xi(u)$ is the positive zero of $f_u(x)=e^x-1-ux$.

(2)

Hildebrand and Tenenbaum state de Bruijn's estimate as $\rho(u)=\left(1+O(1/u)\right) \sqrt{\xi'(u)/(2\pi)}\exp\{\gamma-u\xi(u)+E(\xi(u))\}$, where $E(s)=\int_0^s(e^t-1)t^{-1}\,\mathrm dt$ [2].

(3)

Equivalently, for $u>1$, $\xi(u)=-W_{-1}(-e^{-1/u}/u)-1/u$, with the $-1$ branch of the Lambert $W$ function.

Comments

(4)

The equation $e^x-1=ux$ also has the zero root for every $u$. For $u>1$ this table stores the positive root; at $u=1$ the positive branch meets the zero root.

(5)

For integer $u=k-1$, the same equation gives the $r=2$ rows $\lambda_{k,2}$ in the table of core-threshold minimisers.

(6)

The retained integer family is selected by hypergraph edge size [4]. For smooth-number asymptotics $u=\log x/\log y$, and the same root is the saddle parameter in [5]. The additional half-integer ratios correspond to $y=x^{2/3},x^{2/5},x^{2/7},x^{2/9}$; this is a short exploratory family of simple power thresholds, not an assertion of separate applications for every fraction. The positive root is chosen for $u>1$, not the ever-present zero solution.

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)

Links

[3]

Wikipedia: Dickman function

[4]

Achlioptas and Molloy, The solution space geometry of random linear equations, equation (1)

[5]

Hildebrand and Tenenbaum: Integers without large prime factors, section 2

Similar tables

Minimisers in the core threshold formula of random $k$-uniform hypergraphs — stores the same positive root at the integer arguments $u=k-1$ for its $r=2$ rows

Values of the Lambert $W$ function — gives the lower real branch $W_{-1}$ appearing in the closed form $\xi(u)=-W_{-1}(-e^{-1/u}/u)-1/u$

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

Data properties

Entries are of type: real number

How well the digits are known: proven

Sources of data: [2]

Table is complete: no (it preserves $u=1,\ldots,12$, including the hypergraph family $u=k-1$ for $3\leq k\leq13$, and adds $u=3/2,5/2,7/2,9/2$ as exploratory nonintegral smoothness-ratio references)

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 measurements are in coverage-checks.json. Older attachments are retained for provenance; their previous enumerations are historical, and the active enumeration is selection.json.