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

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Numbers

$u$

$\xi(u)$

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

5/4:

0.4308422097842590367678406745433500325966073944549095698040809928467799856206152855141669049881217428

4/3:

0.5502007416590685881532651793745837564294267158851963986462070093780453807482604711689703852957204317

3/2:

0.7626885608503389820439021764368102222879709347529379724068165212571283274895766670730823743109837147

5/3:

0.9474048816151515968613313591661199195588286321408249292797149755041885517824058814207467101113414701

7/4:

1.031362013654549559322624616872196193473552631135034701705930030878971382796489069315759935323143216

1.256431208626169676982737616609216326916416831701323711125894727048300478541051903533664750947250849

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

9/4:

1.449694645340100687086833176791087767675540734091666708858458754987758795229257956626596402521281420

7/3:

1.508448568483641598987405242582573260390832040224303201117455584174322799352578834091966275801323932

5/2:

1.618788125264683184559765827258905623048980981186977513366824128381904573130926940976004047610963874

8/3:

1.720732284408317659011873338940705399046716204781241642007447230761176288997978661118240883717717715

11/4:

1.768924410810562902095228747567778835488298628123440120204406284747924650688478224243702332948323782

1.903813694440383484710140360828135127280364804561987445879389818281945875445251742935897221209070264

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

13/4:

2.026186682200931369111268138401460067573834677706578467810305401752042945186954878612155755965199723

10/3:

2.064568252537788511254679478828927678048261281988803041290389925256467725855089711098705403152449627

7/2:

2.138109224898635763921094054319228832184279393623334212884171400615023038106414735586297204460649494

11/3:

2.207723149804488396757805953955178344534807864078417947160950945660839242438918228318639994829444177

15/4:

2.241180537530880984993853082284277385942790924155824384018323084173068140532399455522579126200788301

2.336662982263053881182850266714011154679504739557109039685225120116767568104385781543593285556392922

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

17/4:

2.425569854583902484348622765257198460019457407027247428706538852233018222270238681661515479476270611

13/3:

2.453891390863051147145234916953402681632844491083578522526072007884481422199233193443480730656987627

9/2:

2.508726441467471078712734347455562125731353107775127114612782791600653683069932287325654123689115897

14/3:

2.561310682991859721793339968678621237887606079156652324625116606145126985626304865455571184012905389

19/4:

2.586813527765277980939517193996337442817334505056666688860754522380835554096112790775054746535559578

2.660399058463684990432673112693838761316192793557399779133621108141380344526679670369701126317582048

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

21/4:

2.729961615041201614604214528558157514048905828486506625255144043239868558262088408053417943052687452

16/3:

2.752327257111547933106242427420624531393528331274864956963741782636269514407315156990396582185937308

11/2:

2.795908113325161789892457000241883896303096987575879502188931146102248080808432725646163132084600730

17/3:

2.838037825379848718983303330275315872402585924585139685196073153899066738630196947878749574516639564

23/4:

2.858587344627400155635869117390003629432620264567936243344306124888646880709608455342723473915766369

2.918300475783052591304221719797499459970061873861584609944495208244877889764383735414949103879265396

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

25/4:

2.975309290094253365669626529700808913572787101540936313397583442347514217152295712602904337790920610

19/3:

2.993751678247852684240727126142838350263728395931707042132194283071907473159349986448349967334239630

13/2:

3.029842726719178661971253698663791652498899469734803266062238817499714200654287287990888468652318370

20/3:

3.064923569061669171535990021675417268228148755042395279417047759229602556474346817135456864009940138

27/4:

3.082102122196041386682163694820059424568051799967041759273749583272293908250701690278721111856421075

3.132265450540420106833422460869303332477500804253648991520139627783433414358227583586491785309904738

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

29/4:

3.180490782423922825878324760219110223654329057995990284812832112143966544088787887602643122634224549

22/3:

3.196160043458870025716598591955476760732122781449463920940785822316607724840007051810520545709695599

15/2:

3.226919129418414821853700340902534383920819248968167322024952616618420678179070027219718808211794954

23/3:

3.256935953722023658127524471204370300153229644662890456338398537883108073302955365409599354186331754

31/4:

3.271676799623395756570719045320023805229584083004487718558293442004262360717435899272528918071626932

3.314877361786054930901651570191749329550314788552505601434533572470055998782613615635980490671720637

33/4:

3.356623293274819309914137296804517442718977630218714424827391608307833745922495648697286862719809404

25/3:

3.370231643790833789413983513656400982906690340242801436363873014644632404004905965493174425934551417

17/2:

3.397007370903744338233729146943562091548905209566956304953242086703443202186603516227996631111316500

26/3:

3.423215431220685601411800413740866110495510856892117820999772323071051691172885093116283207145323011

35/4:

3.436113851173699788289708250909316791972186210338236648764318629432236216512843577741066611930971779

3.474019476966354505452070987403288659224133427022420159376911107356917224200938678254396618806431641

37/4:

3.510794340359186328691879129557654755275375462376379731138125742857401037835783256970149059804022610

28/3:

3.522812598549406269940406642905875536401600954355834640281709172073055982752412562343809551028623002

19/2:

3.546502625488173711508694735911126152530762368735507433646652602196738976924345013262989078625298620

29/3:

3.569744900351265903613674184041640942263676827583562652744980109076660154987656135719381839791913452

39/4:

3.581203250877008678090857499309220781548974827091979633272223703554472766575206704944634857633088247

10:

3.614950427087530629681888723572661435176889879499709780542263575885132076037722882450886931283670500

41/4:

3.647794142709314010141596688643774798924395495004892136345681422627946046972061956228527660480276656

31/3:

3.658549364605642303557701554686085247898614509467693551214290050921628207082565449154215413480893075

21/2:

3.679780589434310917968767391549419515602495688791834640019766689667761563590615415012438599435900208

32/3:

3.700649877402820900942506710266433671250778422126546950476130011089679671891635580890698771890510040

43/4:

3.710952535131899258085725398958049005237017880504952819990898292124834102309378037854895638163572600

11:

3.741349652356515880952925500625653961859467356437047879191127422890009883669962272286344364902196782

45/4:

3.771008808511908900167761801897846217264989994056321571815287568168046075406353991732574770227426391

34/3:

3.780737203156954922266225917172470943451268435204571866927770916966651051396989149731211403692601848

23/2:

3.799964322906919596207859863326708917880519141780136460342050044054844210617445330715401400228052508

35/3:

3.818892978514570346375852251055043079083921109648162278332278568898706870953149480964016758789812167

47/4:

3.828248195123976160496323355847052603939816609210980412685482602407034964781525169638735641959726289

12:

3.855890308448959712952366737654264389572596946850658243654505887045967065647643973726513680199805710

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.

Programs

(P1)

Python

import mpmath

mpmath.mp.dps = 110
u = mpmath.mpf("5")
(-mpmath.lambertw(-mpmath.e**(-1/u)/u, -1) - 1/u).real

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

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 holds every rational $u=a/b$ in lowest terms with $1\leq u\leq12$ and $1\leq b\leq4$ (67 entries). This includes the branch endpoint, every integer argument and the simplest half-, third- and quarter-integer arguments. The denominator bound selects simple rationals rather than a decimal interpolation mesh; other rational arguments and $u>12$ are not included.)

How they were obtained:

Each entry with $u>1$ is enclosed by bisection in Sage ball arithmetic, using numberdb.bits(digits, losing=64) bits. The endpoints of the final bracket are checked in ball arithmetic to give opposite signs for $e^x-1-ux$, and the stored value is the resulting interval. The row $u=1$ is exact.

The generator checks every stored row against the Lambert-$W$ formula in (3), and it checks the integer arguments $u=2,\ldots,7$ against the core-threshold minimiser table, whose $r=2$ rows satisfy the same equation.