Values of the linear sieve function $F(s)$

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Numbers

$s$

$F(s)$

3.562144835980395970473008206214359098339290428606860410715331753025682153627176587415148432976836561

7/6:

3.053267002268910831834007033898022084290820367377308923470284359736298988823294217784412942551574195

6/5:

2.968454029983663308727506838511965915282742023839050342262776460854735128022647156179290360814030467

5/4:

2.849715868784316776378406564971487278671432342885488328572265402420545722901741269932118746381469249

4/3:

2.671608626985296977854756154660769323754467821455145308036498814769261615220382440561361324732627421

7/5:

2.544389168557425693195005861581685070242350306147757436225236966446915824019411848153677452126311829

3/2:

2.374763223986930646982005470809572732226193619071240273810221168683788102418117724943432288651224374

8/5:

2.226340522487747481545630128883974436462056517879287756697082345641051346016985367134467770610522850

5/3:

2.137286901588237582283804923728615459003574257164116246429199051815409292176305952449089059786101936

7/4:

2.035511334845940554556004689265348056193880244918205948980189573157532659215529478522941961701049463

9/5:

1.978969353322442205818337892341310610188494682559366894841850973903156752015098104119526907209353645

11/6:

1.942988092352943256621640839753286780912340233785560224026544592559462992887550865862808236169183579

1.781072417990197985236504103107179549169645214303430205357665876512841076813588293707574216488418280

13/6:

1.644066847375567370987542249022011891541210967049320189560922347550314840135619963422376199835463028

11/5:

1.619156743627452713851367366461072317426950194821300186688787160466219160739625721552340196807652982

9/4:

1.583175482657953764654670313873048488150795746047493515873480779122525401612078483295621525767482916

7/3:

1.526633501134455415917003516949011042145410183688654461735142179868149494411647108892206471275787097

12/5:

1.484227014991831654363753419255982957641371011919525171131388230427367564011323578089645180407015234

5/2:

1.424857934392158388189203282485743639335716171442744164286132701210272861450870634966059373190734624

13/5:

1.370055706146306142489618540851676576284342472541100157967435289625262366779683302851980166529552523

8/3:

1.335804313492648488927378077330384661877233910727572654018249407384630807610191220280680662366313710

11/4:

1.295325394901962171081093893168857853941560155857040149351029728372975328591700577241872157446122386

14/5:

1.272194584278712846597502930790842535121175153073878718112618483223457912009705924076838726063155915

17/6:

1.257227589169551518990473484546244387649161327743597792017175912832593701280179972028875917521236433

1.187381611993465323491002735404786366113096809535620136905110584341894051209058862471716144325612187

19/6:

1.131913324960796836604751463090615454287264968173931419949463140997188665916843041330570629002615706

16/5:

1.122982298588715110920438537943982689569175608082952584779502526357768914441645065924123081349529578

13/4:

1.110700741838247928393123053329078130278523009466266687432022947053849151936301849184076557823128208

10/3:

1.092865486186890039912897923104294787785632678413750706672840442204001180882367204592760507050926650

17/5:

1.080649975555578875961114468661412866664417331726471771514100716946641299989760478256571270496002084

7/2:

1.065193558002913994269316518452595346668643921351336197420530443602274381297809956462586587618990693

18/5:

1.052597856458764913296283780867563300994690773104556487123141893087691823682683327901947925457491036

11/3:

1.045523297907216096026810457370852323146774360893472228885885010859986161103695153987178141836391833

15/4:

1.037936175776293353696364245785020799229799792609447221998776548136923504437758527863827435147029995

19/5:

1.033969718514457784772991185316165258281463619881208707775510455321458885492541052584192017316811980

23/6:

1.031543707628241583186776356915812703800689233975347881819813659665521981539422529970146257754279999

1.021641552540073820678763132674147481781985707438914026027553485555437809356538234848319766504667403

25/6:

1.014673049030001662204824250720469135080244829884656936868627066779094159161511959927561874454386781

21/5:

1.013554653251099026111088700491512668624126436081722504923725779951894453117968560071763276534912112

17/4:

1.012022559835273789036819123262822704171461918331712111910524298069506614541745328512573530515934950

13/3:

1.009816875798388580069980516803835651022134989932809713495716776024322034364431309007839242345746569

22/5:

1.008326441525278614915490089016356018765283378298328303534662774420929635183146223974461868197579304

9/2:

1.006477127645245870545538534107507916037032273903400852146364364829083911704260393877625197439668209

23/5:

1.005014608064518841654188024755913096604370498450451548553996851794000407619857040337847983160619387

14/3:

1.004217854737140914848050351482015545907911734928985282957766816888744778241055657220796520242319976

19/4:

1.003389530765110444130040780248842231831290733436123181531450227885528574266942959242762984958244087

24/5:

1.002969675439035039218935053611915115627771286293798272597734069603504900698384510461868581213343636

29/6:

1.002718083681778275191567098717137120686380208575786963868321011832623658821400304522845699393631272

1.001740410233906606991474539965273713399186684278221039455252367542175224469344275472319557164798962

31/6:

1.001109567677161695854337614371749317538365271418309271020916674761225655189651792825905190741665227

26/5:

1.001013419971633189041614341729116464154658384092297458912402663718560016583152677214839241929387949

21/4:

1.000884208906670278840703763087133208371950779800068666753082647816304556420311219966287299214501606

16/3:

1.000703547143587804605684766205106469002923094659703676663270988238934747343272743194343967703435693

27/5:

1.000585281478898894106376020536867338398457371657440113322116191899886208960677875084804406381574482

11/2:

1.000443141619517211257244697864465587283974538112659371403578152926221490948785814470862824844240630

28/5:

1.000334608640742560382237700359920323611876624905892978208732804091422280226292037222781728793810431

17/3:

1.000277021761621651146000795791454335526431194594280735149357395524812252079634389636188654903488613

23/4:

1.000218369648926811566055772657184940507256880512369561761978405310273774652498945635934992535725949

29/5:

1.000189135011376299179489039658497781725698057558235650589511836453214057775452860734611469405711375

35/6:

1.000171782846378711150760287407393267507681340237395293696521564383598301631188818095475359360067562

1.000105656810419070324353819877366508417832493913975973185105156669515971789770680079274728565623455

37/6:

1.000064479592945584409449690467608416967977581517469463549723574400214341326248176747602520821580712

31/5:

1.000058364262013741485806482604360419498893209017729986960900463333193005790988367837122733360615673

25/4:

1.000050235586508291573838257928894497927793523407498071563020005843938635701459775423900337567678115

19/3:

1.000039074078184976731091819652149497915058277326417138280496472788847175646576670434561646729948560

32/5:

1.000031923950454995723296565979518865982167205672996599369681474512265146443855094188596782228098935

13/2:

1.000023536154978957900067437385514183760315563498850536658621001148520993167530979239234889424047593

33/5:

1.000017320488724106150239306108479182621628796397791583728711159110681503156745774699603110395417245

20/3:

1.000014104726930873503571356406813981699822997388660463911316199898113160131821521911168467925484843

27/4:

1.000010899871044299159938669491578865107429650548640087248435199211728708727673713536033307473068673

34/5:

1.000009332872636560668146334249999786246720714114885773329199488317283789470974564809216413855465154

41/6:

1.000008413488286026689655298224364549991364643922616741443292191798639263798662353659745603764253671

1.000004994522598344765396414866596003357933575845658272394226961334011903575671961139562583970127255

43/6:

1.000002949195741273521093658779263058583575339819037084334947266380147171481743719952853939989134388

36/5:

1.000002652485149939989139151309987830061233364530359449467707497007320390348751482347308868516327337

29/4:

1.000002261466119906402916179367508156375221783036760898634091415336656994034960872969815366595809237

22/3:

1.000001731609422543809059187429438175817872632108943844774429724624886227703894719152535124423530246

37/5:

1.000001397155823636346586691404543333693948600555019829948212986672619659447380630911901191081478079

15/2:

1.000001010850003507042903346221434506267750303635796142301314678511627264997907906896745011034023861

38/5:

1.000000729866165565517709537128050964582607218957563005945849751397411817801185779640445549591190531

23/3:

1.000000586770908882228945639316632781912622119517114096699248383896332313044284248196064319590100611

31/4:

1.000000446147964810227455358898226082594365821469222379424595386457430223458197305088064244349678197

39/5:

1.000000378281948958582636233859515910490183453275695752304358015145691290847708585488254641787556315

47/6:

1.000000338790279812412271481204550591473895291080935870091874569864158665061289273770626284868283102

1.000000194650550873634996312222899830569479369066727640917743055975324589197507457955217692290426708

49/6:

1.000000111333090034058437633340964229595255815919087176541993449922465790795714295357825996017775689

41/5:

1.000000099512184959678718796412714370286244689735765228332167453107185214296528321621294929796380995

33/4:

1.000000084065832455578271111016673962107879476727882801790456505909217467643767629566242777933781002

25/3:

1.000000063411711994752624783593462402726382511185459624997943093732054103959198068781440303206718801

42/5:

1.000000050569814363504514977207873584814035028952176509407434863276134584644008725099791135519392189

17/2:

1.000000035970134532629910217946836914344841163799548848876911625680041798317291176978901998964618239

43/5:

1.000000025547286198608102720469891458307343796248955334041512903313557762056013180056036823547381164

26/3:

1.000000020319472110345696510811153439413337612055206436681029180776801110621153219626546396856520392

35/4:

1.000000015247549219293976478835025982269907586674065445101884027760648962420340821357646290664009242

44/5:

1.000000012827570142694730202049934907708796809993718342200405742884392523829170551118123803962275672

53/6:

1.000000011429081984619298034986843260792691743085240259715349040609577378501859867775569043825343368

1.000000006400114006046968733723925275130075753569090788163467947102711857740561311833631549252893010

55/6:

1.000000003568155033015911404252109311041994862815357288160125818831359594042932693170756635461350143

46/5:

1.000000003172988588527164835179919418748462418729184876062475227756480924393900129399410393405807192

37/4:

1.000000002659909924066620455406341574255153417121226065779432702636993128832957978333331765774763961

28/3:

1.000000001980767235566442959644916638069092794888284242137922227740766726813689028175659896161720156

47/5:

1.000000001563456189642148343995176368214560057405284241255151857658253749418770176313523643348530741

19/2:

1.000000001095075590537413350344997726164144743953261531337977010037586736831949370940871631340880350

48/5:

1.000000000765944457440728859317145293162403258894197487533686282380199670294461424901580500852581024

29/3:

1.000000000603077205349739205500452521383242156474919798558196226926129974309196866960423152092861128

39/4:

1.000000000446924378466268452751943201658448070456412001321753548523743053501580133780920881569332090

49/5:

1.000000000373218086650146066758719622037990759308545534580704562224196497244346199644776793587825846

59/6:

1.000000000330905735245660319248528273006158256346888536314752932045036417090855469666637417900271303

10:

1.000000000180922500032806009733696234663186629562323214970635089055242817978172010850762225377460680

Definition

The upper linear sieve function $F$ is the upper function in the dimension-one linear sieve normalisation of Halberstam-Richert and Iwaniec [1]. This table gives $F(s)$ at rational sifting ratios $s$.

Parameters

$s$

— sifting ratio ($s>0$)

Formulas

(1)

$F(s)=2e^\gamma/s$ for $0<s\leq3$ [1].

(2)

The coupled delay equations are $(sF(s))'=f(s-1)$ for $s>3$ and $(sf(s))'=F(s-1)$ for $s>2$ [1].

(3)

$F(s)+f(s)=2e^\gamma\omega(s)$, where $\omega$ is the Buchstab function [2].

Comments

(4)

The table uses the normalisation for which $F(s)\to1$ as $s\to\infty$. In the sieve, $s$ is the ratio $\log D/\log z$ between the level $D$ and the sifting bound $z$.

(5)

The entries start at $s=1.00$. For $0<s<1$ the linear sieve gives no sharper upper bound than the endpoint value $F(1)$, and the family convention keeps the stored rows to the range used as a sifting ratio.

Programs

(P1)

Sage

import numberdb.sage as numberdb
from sage.rings.rational_field import QQ

# The attached generate.py evaluates F(s) by the linear-sieve
# delay equations and can be run with sage -python generate.py.
from generate import LinearSieveF

LinearSieveF().value({"s": str(QQ(4))}, 100)

Links

[1]

Mangerel, Weighted sieves with switching, §2

[2]

Tao, Chen's theorem notes: the linear sieve

[3]

Wikipedia: Sieve theory

Similar tables

Twin prime constant $C_2$ — an arithmetic Euler-product factor in prime-pair asymptotics, where $F$ is an analytic sieve factor

Hardy-Littlewood singular series of prime tuples — stores singular-series factors for prime-tuple sieves

Bateman-Horn constants of monic quadratic polynomials — stores arithmetic constants for a family of sieve-theoretic prime-value conjectures

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 holds $F(s)$ at every rational $s=a/b$ in lowest terms with $b\leq6$ and $1\leq s\leq10$, which are the arguments a reader is likely to have written down rather than the ones base ten makes short; on $0<s\leq3$ the function is exactly $2e^{\gamma}/s$, so every entry there is a closed form)

How they were obtained:

Each value was computed in Sage real ball arithmetic by solving the coupled delay equations for $F$ and $f$ on successive unit intervals. On each interval the functions are represented by Taylor series with ball coefficients and an explicit geometric tail bound for the truncated product by $1/s$.

The entries were checked against the closed form in (1), the delay equations in (2), the Buchstab relation in (3), and the formula for $f(s)$ on $2\leq s\leq4$ stated with [1].