Values of the Buchstab function $\omega(u)$

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Numbers

$u$

$\omega(u)$

5/4:

4/5

4/3:

3/4

3/2:

2/3

5/3:

3/5

7/4:

4/7

1/2

9/4:

0.5436193561396487803405755956932597792776004824657809838539057211055074416747859259637365440445965492

7/3:

0.5518637453364775403310938597116403277872184475276118813599995794354112645946201990020475282196165513

5/2:

0.5621860432432657527912052461857396546287961693849976790456057296576402684995657005071009711269253605

8/3:

0.5665596089122465062020677861138732255792915486671631013167325841887567604333932991586942962297928208

11/4:

0.5671330137946991586439594547370278521767579857677638837156473203457427704773381223912479434891033280

0.5643823935199817698057440404860588560251667114534184180402266698311312073232315718686211089988062292

13/4:

0.5613237220599537001444931453784749130687257311010794387176112115606231540744649394181401953321648125

10/3:

0.5609892038574431640727047689154029418725909316267629713268627526962168806099422927874900746002709867

7/2:

0.5608288644515888217081409622410368236057816049907761493582358191472449403154531431437978766645294358

11/3:

0.5610083726350823381822566761954286245998054016043401471639908180525408101599839845962350848365540148

15/4:

0.5611397977276340210411258038832953383471470283860567009777397040010243445070194811344428797770435894

0.5614582414068377374244183782971147076623276740309098614465484880816192055432876014378307007496579714

17/4:

0.5615209072567544792844398427414221043095751609716931088211645350615607207192528947033839148032206892

13/3:

0.5615135373678223600859648997094023845709031497765132156599421360116895842180741189930414851428369157

9/2:

0.5614895520333070465067980893790981790128752825592844902745012395489992052926495568949780329846570383

14/3:

0.5614686095077393733256379958827675625471794403530870305283716849906497612295195280290682799336297854

19/4:

0.5614616779071430203755767946536208889755312077589785769566353616506402466115670563956647908196933386

0.5614544682684567287363246889425285326620983483162800302968678707644020228811670126648449201606215930

21/4:

0.5614568922439256374194366829325981102104804330245791430194922885973432231129373053973443630901040113

16/3:

0.5614578429552539961689913464918515663034800632097536752885994584729954333820552572480480207460498523

11/2:

0.5614591773151575495924972793161599395334203183069915787020547392288848987450930281796559220397209900

17/3:

0.5614597426133725254854978063608313326843297934160874396061069856558476580222485790476047078566213965

23/4:

0.5614598177769629430694499620604391527553016257901012015588161090702685129666788534465363625559213082

0.5614596848016890453857407290415545292990093348345936324170409534788144194274186385747928001461618182

25/4:

0.5614595159327636942837028641033449078024223462494013934741066651129913733505506598957801219979563100

19/3:

0.5614594878266521097916353061543890561703732266948792567974728556100340258582235287390063783694381961

13/2:

0.5614594645318247269995811134018116561801035502638870044850206774902057188244525369390270847892064163

20/3:

0.5614594657966931645055501711529312063496370549654563262226833656243621207580190283053777027168131817

27/4:

0.5614594697845266287435257621232379393916711731354723863290787972394729863960157858013650131213626635

0.5614594806894847631436195446529916782578631714436753446494409067676190754966361811635894294979436975

29/4:

0.5614594844739103202857682231535116792028453630677245261900625060975433532510543826834317290100520021

22/3:

0.5614594846590546016769240040466922718628932271584891770040228550504100537051278625554963202320984829

15/2:

0.5614594844041371462874719935972446389864551441523825569884090587809086712743533964460662297165487641

23/3:

0.5614594839602889370933469324649407280149174031514444837174828784112109920416323843066505872790041902

31/4:

0.5614594837856811263730431591025398206073066767453284703144260991886563212380607701399600781553137186

0.5614594835344085231709984685685153333556755808799845705042446865033247303354093400695417269392091758

Definition

The Buchstab function $\omega$ is the continuous function on $[1,\infty)$ with $\omega(u)=1/u$ for $1\leq u\leq2$ and $(u\omega(u))^\prime=\omega(u-1)$ for $u>2$ [1] [2]. This table gives $\omega(u)$ at roughness ratios $u$.

Parameters

$u$

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

Formulas

(1)

$\omega(u)=1/u$ for $1\leq u\leq2$ [1].

(2)

$(u\omega(u))^\prime=\omega(u-1)$ for $u>2$ [1].

(3)

$u\omega(u)=1+\int_1^{u-1}\omega(t)\,dt$ for $u\geq2$ [2].

(4)

$\omega(u)=(1+\log(u-1))/u$ for $2\leq u\leq3$ (3).

(5)

$\lim_{u\to\infty}\omega(u)=e^{-\gamma}$ [1].

Comments

(6)

In rough-number estimates, $u=\log x/\log y$ measures the gap between the counting range $x$ and the excluded prime-factor bound $y$.

(7)

The function tends to $e^{-\gamma}$ as $u\to\infty$ (5), where $\gamma$ is the Euler-Mascheroni constant.

Links

[1]

Wikipedia: Buchstab function

[2]

MathWorld: Buchstab Function

[3]

Tao, Chen's theorem notes: the linear sieve

[4]

Wikipedia: Sieve theory

Similar tables

Values of the linear sieve function $F(s)$ — another delay-differential analytic factor in linear-sieve estimates

Values of $\xi(u)$ in the Dickman-de Bruijn estimate — gives the saddle-point parameter for estimates involving the smooth-number counterpart to rough-number counts

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

Hardy-Littlewood singular series of prime tuples — stores arithmetic singular-series factors that occur beside analytic sieve factors

Twin prime constant $C_2$ — a prime-pair singular-series factor from the same sieve-theoretic neighbourhood

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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 every rational $u=a/b$ in lowest terms with $1\leq u\leq8$ and $1\leq b\leq4$ (43 entries). The integer breakpoints of the delay equation and the half-, third- and quarter-points in each unit interval are all included; values on the initial interval $1\leq u\leq2$ are exact. Other arguments and $u>8$ are not included.)

How they were obtained:

Each value is computed in Sage real ball arithmetic by solving the delay equation for $h(u)=u\omega(u)$ on successive unit intervals. On each interval, $h$ is represented by a midpoint Taylor series with ball coefficients; the product by $1/(u-1)$ is truncated with an explicit geometric tail bound and that tail is carried as a ball radius.

The generator checks the initial interval in (1), the closed form in (4), the integral equation in (3), and the approach to $e^{-\gamma}$ in (5).