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 $0\leq u\leq1$.

7/6:

0.8458493201727416957071246149375225430859059626011435173959841931933080524362344146918205832222792493

6/5:

0.8176784432060453737882819748454853668026106620855130160572735483432910725193540821506547962830288345

5/4:

0.7768564486857902442337049096901654966253989144519927863287121275126082562317316665815927758996577643

4/3:

0.7123179275482190725607809940061725684964902891022389434933343146507070492792195356618891008208947137

7/5:

0.6635277633787870694954065897830079098885166246866565334532577415365991249555884968424753795053080838

3/2:

0.5945348918918356180219868845356508634280095765375058023859856758558993287510857487322475721826865988

8/5:

0.5299963707542644463490629688516579352991009511877519595506078629939981217985736178125438971039235482

5/3:

0.4891743762340093167944859036963380651218892035542317298220464421633153055109512022434818767205524780

7/4:

0.4403842120645773137291114994731734065139155391386493197819698690492073811873201634240681554049658481

9/5:

0.4122133350978809918102688593811362302306202386230188184432592241991904012704398308829023684657154332

11/6:

0.3938641964296844567505337804155729729012838382455875306368066340003142912750669694044243738075159849

0.3068528194400546905827678785418234319244998656397447458793199905066063780303052843941366730035813125

13/6:

0.2330556073185459670340125732011339348671590367814520882815155879952832980820626093722371578058865921

11/5:

0.2203571379083276335094518264713741948384416877356977073111033017384815375777643614281919810625126549

9/4:

0.2024416642621919975575502606226469211853969322285313853761791409956225657463540480379459235602820356

7/3:

0.1753682920837366528221342166767017562627814285709864309734842097205517962995537074635480944526244127

12/5:

0.1559912638725041416104691577878684427996594339891610961189212930960410133055379350553922141071908480

5/2:

0.1303195618322507456114389443076067397200331776511915871698896030537865596281599029547514661323311788

13/5:

0.1082724429762708911227211903062777148250995884440041927555031482619695673572975499921640152890285342

8/3:

0.09537219363851609962203889114589695770793344869988125167994160984060982520565758363158019204965010643

11/4:

0.08107241812166772844926309446480519146038818601389929471210850971626853195362849052750469114431862745

14/5:

0.07339158076259944121858630488698400344987482686604000840506299780493547416525226313032459919289274492

17/6:

0.06862195344828205586504718065395030544161524913797026897326388799982139861031093580594047567293553027

0.04860838829113156690718303934340742135432958047814054231680528505148823573593247200409129337116770797

19/6:

0.03407692647239495873163361367392522348130947200653989810731317198527245643716353543036793273424269667

16/5:

0.03170344451178008620866706728899487898589932587438721053396070750205108734430017778742611172008494736

13/4:

0.02842721532218082472125479306602439089521911807194077971136902611585986963826898896505609893590757019

10/3:

0.02365013826528736691310475020842872579947483447727156796198981771320042290822285015983532941585991464

17/5:

0.02037177906040766832088276803226748390856868863050093947720106969792381074804495459998713262808686149

7/2:

0.01622959324323599163094189604678439986680615911030965241451229416257007039186053275168936549155862219

18/5:

0.01287543418667647230195129891840279650526327376802432412381209829158514212339700337853984886764266087

11/3:

0.01100880014089834881558267389033129182907566123089389685963712138098401759178165796108958916347416842

15/4:

0.009029226800111860712835644554425695407981151006681065681272710757619450041936748958381873905831077208

19/5:

0.008006872188385230650719713564759833462190465914719545510105900112230917056717377196245844211387851385

23/6:

0.007386890617100897812678718200764427804355375159400983207191385470193425274715409214608472459174057985

0.004910925647760832352739150923615186032484297417692945977961657528030631494153505349840165957891479305

25/6:

0.003237457130392909664196827245170415226207596255220932329713915961910970362774391142771688787975550616

21/5:

0.002975474789581521478314609047887193872402704717835873742977953216305534429130409248416150239556589667

17/4:

0.002619953695085303484031462044621715365015218412548431476141574096454029034253253231990077576262797936

13/3:

0.002115487080186344726062503884960812036559233539851844306222989968801408269303023069769692354148751165

22/5:

0.001779942464815349752063626713662726801482535091145229424901702660768476651436674961065458856023599421

9/2:

0.001370117741128107322870660366958231234165813445148768449972749640039348035902946329044509035023296460

23/5:

0.001051444855432390722183195805097017856154395479259491977504766219799086864770260227570579394721236948

14/3:

0.0008799055428006437127143927638596570416779566655575918901656091969537741470075479620689511374589426310

19/4:

0.0007030611263532992823225524948041890066395359503653282965455543054322312326743701701875740929800797451

24/5:

0.0006139573219700945169573107394222985222748054275154738087135005738212141058655696259902773365669833645

29/6:

0.0005607161655104877644096807045177381236021370693651830407091512487209246396425069934722512327734621918

0.0003547247004560397298338945107706235609516436105726162701669036821477003206922126803906929382184528416

31/6:

0.0002227885260864938681331850939550419397161679895636615193659680136521288560754297220783204265669433952

26/5:

0.0002028215348055161201711798366506744038931239084077948180167787922730682915707676082341796606210872708

21/4:

0.0001760805036193781980612726025753760722000365708660686206687543651418282907601739828670062230185949093

16/3:

0.0001389169292804446911553060160238420526277031371085711121298831882046213603336716785763049886536723886

27/5:

0.0001147741966215643972478947610250453276249749917985186628796926333795285828816044233212259250873194846

11/2:

0.00008601861112051155146239828622280250925731074119364361091142281064157539966809547539978039396825211140

28/5:

0.00006431468046151080711480082032378678645359775509935107823340877425052277377577563394933462318331045619

17/3:

0.00005291335260310328778764968076216227504221946699687016866757425261871579542815004452173275244697817171

23/4:

0.00004140192370062779303823771271358800492405636711283172168385039916530846099467432383681010654184445608

29/5:

0.00003570834903825215606397938852234697333414087183643300107700134820616713668263897598287839524620004847

35/6:

0.00003234459231595809959859090765834431139934787131078734603216426675555558725706132809817925161652510066

0.00001964969635395528965175498612920452289459671980962327267045628854204220362614428655654864830520116536

37/6:

0.00001186473802221636590648580128164119976629010203543471849299007333029646261788411861788798805000328277

31/5:

0.00001071830445086802264597109489665904868519427786099538138453595069359415105455306222503704861337482958

25/4:

0.000009198905666112413533472114981429767716106105889654499975662361737255520630612888423136550550614320709

19/3:

0.000007121666659720407666671568894771876259914378574094339652433768428371592145934760719396568744367894610

32/5:

0.000005797105944950749618520249797918718744764488822952124795102664222541118061268353529555574506372893263

13/2:

0.000004250355517171387818732059049633956745560630444671349428819436699515972909963466809202016338628286837

33/5:

0.000003110126499791365043715833514510481506745557922998211979441489427652233280207118619771141242709879196

20/3:

0.000002522752605744611523106054190052435788648840265172826979041746582127144212872150674957681782003212113

27/4:

0.000001939632877191691554153329997966603409759827812106452339324391470858593208788325494032964918500377903

34/5:

0.000001655570663799224832892933299252240329039530522351061952403514846948047675072754030997946173720919455

41/6:

0.000001489307193689578823257067110626690489563907781784560344674023204489860804714342759642401451020754427

8.745669953293916695580283572769972173380471976458048319843283490701751507455205957369890872419066466e-7

43/6:

5.109205051286917077938359074423919036279922084994559173085742083296475941112556523281350900558601017e-7

36/5:

4.585655128044050527694699557675054771771154277352274831128552964191639892464154794054330096825206697e-7

29/4:

3.897723683911094692235537062557913006762279294986932178934576732996045323075793908291958166282680572e-7

22/3:

2.969830098369656820003589853404328832757874389521294888092298720591554924383097502107696253399437849e-7

37/5:

2.387186129813229073463295541218996430064411792153807711385461774877137831770955517446985841292664940e-7

15/2:

1.717867492033985818036125909629327176109062948477731761555795402466024407930773511126785715187322998e-7

38/5:

1.234090210805017248333927396734335101447725323856474886207152387644142839062277363120173146310419183e-7

23/3:

9.889505850357839789095894689081869972768320752552618303559441021822203989213062202684764787612356489e-8

31/4:

7.490339771991795821715132100491121269390099395414894361559716983600181962461416296649072705492021477e-8

39/5:

6.336587433060612241358921191539076201604501620913665429709531700367440573721771594288173651230383125e-8

47/6:

5.666624920824161638838496465665657930282064985461920860535090145909887034825152277518772365796721923e-8

3.232069304226103772599785361728216157619475162802409072273440324202075270546185324928313771781708734e-8

49/6:

1.835233081548574520494960782690713380119371558420259281490544971381680594090094972749605090473006582e-8

41/5:

1.637963044115808633219543482173411143509414560850495648130633842119806389144079503497734075237549010e-8

33/4:

1.380644228072205987503674111032295303789798928226353232109440101605291247652708970080807746060233258e-8

25/3:

1.037535381057364302199051942978712750792473472821334678228399412079218678359603499658648047432885879e-8

42/5:

8.249069972003646181677304294718443439002230236741225961426273904433069397571229507007151961607702367e-9

17/2:

5.840569562936228038434560159482135396173664095298187404429950809998665875835126632813554022502822312e-9

43/5:

4.129032335576973450997321630108461114642850863840184755922533093764209112932482449201654078882185670e-9

26/3:

3.274026269479054358262714872735718485057646419130477483481889804396259482939204783656833064300918261e-9

35/4:

2.447494538023843082731513268075376340596768705774346183101275420073948303387255583482954278204910077e-9

44/5:

2.054435052933073279524358200747521690898811179678065838150344081192519535684940830515327035881216380e-9

53/6:

1.827755325821917329697966304523719435406731056379300153156481845511019627671243541824294924434994798e-9

1.016248282737836546534853935695695783824439958658074404463809039602732886867963160647959275185158800e-9

55/6:

5.628129927201774805461511645427633284242591410353610430008742929598284384885721304530390942295093332e-10

46/5:

4.998432718682924470185734841322541123635838855036397901503162981685635201809101100170359495210231723e-10

37/4:

4.182275801461823376733370144637204198114141683905768021422805689725993200949062038172778986064574707e-10

28/3:

3.104873433681821734266959572604371413899232948421651852042107488613177352326149272215103101539323049e-10

47/5:

2.444842262276526433490288945635890178630308063082481489247986512521613940588240722180031751206277125e-10

19/2:

1.706352738635339302906295134061510336498043470124371997428795635986504821331942282023589817897206561e-10

48/5:

1.189327378016704194537239520824211774616288077723816233577585060977626992232750160218381108614022183e-10

29/3:

9.342600368964430051593186823723259115473751957025400236620768110268047814581277456944538683602996104e-11

39/4:

6.903459800535792975264072764720774786723924800936334621296338190999690831125226295210098868536268269e-11

49/5:

5.754879560794799059216909415752305031151424371760551518908761282299429786998775612156721766622715492e-11

59/6:

5.096482491443811609910952718236498726928079468485386465292519816048626242777144721025010584691126191e-11

10:

2.770171837725958988758121200634342326343006650115606506625469924608958391571159227442464413125831985e-11

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 $u>1$ [3] [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.

Programs

(P1)

Sage

from sage.all import dickman_rho

dickman_rho(5)

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]

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 holds $\rho(u)$ at every rational $u=a/b$ in lowest terms with $b\leq6$ and $1\leq u\leq10$, which are the arguments a reader is likely to have written down rather than the ones base ten makes short; $\rho(1)=1$ and $\rho(2)=1-\log2$ are the ones with closed forms and both are here)

How they were obtained:

Each entry was computed in Sage real ball arithmetic by the method of steps. On each unit interval the generator represents $\rho$ by a Taylor series about the midpoint with ball coefficients, applies $u\rho'(u)=-\rho(u-1)$ coefficient by coefficient, and adds a geometric bound for the omitted tail of the Taylor series for $1/u$ to the ball radius.

The values on $1<u\leq2$ were checked against (1). Eleven sample entries between $u=3/2$ and $u=10$ were also compared with Sage's independent dickman_rho implementation [4], which returns a 53-bit real and agreed to about 16 significant digits.