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

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Numbers

$s$

$F(s)$

3.562144835980395970473008206214359098339290428606860410715331753025682153627176587415148432976836561

1.781072417990197985236504103107179549169645214303430205357665876512841076813588293707574216488418280

1.187381611993465323491002735404786366113096809535620136905110584341894051209058862471716144325612187

1.001740410233906606991474539965273713399186684278221039455252367542175224469344275472319557164798962

1.000004994522598344765396414866596003357933575845658272394226961334011903575671961139562583970127255

1.000000006400114006046968733723925275130075753569090788163467947102711857740561311833631549252893010

3/2:

2.374763223986930646982005470809572732226193619071240273810221168683788102418117724943432288651224374

5/2:

1.424857934392158388189203282485743639335716171442744164286132701210272861450870634966059373190734624

7/2:

1.065193558002913994269316518452595346668643921351336197420530443602274381297809956462586587618990693

1.021641552540073820678763132674147481781985707438914026027553485555437809356538234848319766504667403

9/2:

1.006477127645245870545538534107507916037032273903400852146364364829083911704260393877625197439668209

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 the right-hand endpoint reference $s=1$. The mathematical continuation $F(s)=2e^\gamma/s$ on $0<s\leq3$ is not a claim of a sharper sieve bound for $s<1$.

(6)

The coupled delay equations in [4] distinguish the initial branch, upper transitions at odd integers, and lower transitions at even integers. The first four half-integer references give interior values rather than only join values; they are a bounded exploratory choice tied to those branches, not an unrestricted half-integer grid. They stop at $9/2$ to emphasize the first correction to the elementary upper branch. All earlier deeper transition values remain.

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))

Links

[1]

Mangerel, Weighted sieves with switching, §2

[2]

Tao, Chen's theorem notes: the linear sieve

[3]

Wikipedia: Sieve theory

[4]

Tao, The linear sieve and Chen theorem, Exercise 1 and equations (8)-(11)

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 preserves the endpoints and upper-sieve transitions $s=1,2,3,5,7,9$, adds $s=4$ at the next lower-sieve transition, and uses $s=3/2,5/2,7/2,9/2$ as exploratory interior references for the initial and first nonconstant branches)

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 delay method uses degree 320 and a separate uniform remainder propagated through integration; the second run raises the degree to 420. Earlier deep delay entries retain their historical independent validation; this run rechecks their enclosures and order agreement without claiming a new independent deep-delay computation. 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.