Some number-theoretical constants

arising as products of rational functions of p over primes

In September 1999, Pieter Moree asked me to help with high-precision calculations of some constants arising in various contexts in elementary and analytic number theory. PARI/GP running on a few 333 and 360MHz UltraSPARC-IIi(tm) CPUs soon made short work of them. We pushed the calculations to just beyond 1000 decimal digits.

The basic reference for our method is

Many of these constants appear with explanations and references on Steve Finch's Favorite Mathematical Constants site (abbreviated FMC in what follows), and are cross-referenced to the corresponding pages there.

We regard the constants as given in the form of an Euler-type product over rational terms 1-f/g with rational coefficients, where the degree of the polynomial g is at least 2 plus that of f, evaluated at all primes p, or sometimes at almost all primes (e.g. when one factor would vanish for p=2).

Products of terms of the shape 1+f/g are readily accommodated by moving the sign into f. What really counts for the computation, however, is the behavior of g-f and of g.

In its original form, the Euler product converges abysmally slow. It has been folklore knowledge for some time that it can be transformed into a product of powers of values of the Riemann zeta function

prodk>1 zeta(k)-e[k],

however, although the convergence tends to be a lot better due to the exponential convergence of zeta(k) to 1 as k increases, it is still unsatisfactory, and this product does not converge at all when the exponents e[k] grow too fast.

The trick which makes these computations feasible is to compute the contributions from the small and larger primes separately. By choosing appropriate cutoff points, we could obtain the desired 1000-digit accuracy using not much more than (typically) 20 or 30 minutes CPU time on a 1999-vintage Sun workstation. If several computations of this type with similar accuracy requirements are to be executed, one can save time by pre-computing the zeta(k) just once to the maximum required precision (and maximum required k). For a target precision of 1000 digits, this step takes 15 to 20 minutes; so the gain is considerable. Therefore, we usually ran batches of three or four or five computations sharing an array of pre-computed zeta values.

There are some hidden duplications among our constants, in the following sense: One can introduce or remove factors of p2/(p2-1) in (g-f)/g, and compensate for this by writing a suitable power of 6/pi2 (inverse of zeta(2)) in front of the product. It is not obvious which rational function within such a family should be preferred. One can use this transformation to ensure that the degree of g exceeds that of f by 3 or more, but this does not always lead to the simplest or most natural shape for the numerator and denominator, and it does not help to improve convergence or to make the computation faster.

The inverse of any constant of this kind is another constant of this kind, with g and g-f swapping rôles.

The Tables

This table provides an overview.

Table entries are structured as follows: entry number; numerator f, denominator g and starting prime p0 in the original formula

prodp (1 - f(p)/g(p)),      p >= p0;

approximate value hyperlinked to the corresponding entry in the full-precision table; and a reference hint about the context in which the constant arises. Entries in which f is `positive' (in an obvious sense) come first, followed by entries with `negative' f, and in each group we begin with entries where f and g are of simple shape: often the numerator is 1, and g is a product of factors p and (p1), before proceeding to other candidates. The exception to this rule is the infinite sequence of Hardy-Littlewood constants of which the Twin Prime Constant is the first member, and which therefore appear immediately after it. The numbering is otherwise arbitrary.

# f g p0 valueReference
01 1 p(p-1) 2 0.37395581361920228805 Artin's Constant
02 1 p2(p-1) 2 0.69750135849636590328 Rank 2 Artin Constant
03 1 p3(p-1) 2 0.85654044485354217443  
04 1 p4(p-1) 2 0.93126518416000433439  
05 1 (p-1)2 3 0.66016181584686957393 Twin Prime Constant
06 3p-1 (p-1)3 5 0.63516635460427120721,
2.85824859571922043243
Hardy-Littlewood Constants C3 and D
07 6p2-4p+1 (p-1)4 5 0.30749487875832709312,
4.15118086323741575717
Hardy-Littlewood Constants C4 and E
08 10p3-10p2+5p-1 (p-1)5 7 0.40987488508823647448 Hardy-Littlewood Constant C5
10 1
2p-1
p(p+1)
p3
2 0.70444220099916559274,
0.42824950567709444022
"Carefree" Constant
11 1 p2(p+1) 2 0.88151383972517077693 Quad. class numbers
12 1 p3(p+1) 2 0.94773326214367537594  
13 1 p4(p+1) 2 0.97582415304766824168  
14 1 p5(p+1) 2 0.98850439774124690875  
15 1 p2-1 2 0.53071182047204479497 see 25 below
16 1 p(p2-1) 2 0.78816250003022070058  
17 1 p2(p2-1) 2 0.90192603958708217138  
20 1
3p-2
(p+1)2
p3
2 0.77588351000389549962,
0.28674742843447873411
"Strongly Carefree" Constant
25 2 p2 2 0.32263409893924467058,
0.66131704946962233529
Feller-Tornier Constant
30 p+2 p3 3 0.72364840229820000941,
0.74397119335037474469
Sarnak's Constant
32 p p3-1 2 0.57595996889294543964 Stephens' Constant
40 1 p2-p-1 3 0.71546823598995584509  
41 1 p2+p-1 2 0.66958029053906236764  
42 1 p2-2 2 0.38894518997956192931  
45 p9 - (p-1)7(p2+7p+1) p9 2 0.00131764115485317811 Heath-Brown and Moroz's Constant
52 -1 p2(p-1) 2 1.33978415357434724660,
2.20385659643785978787
Totient Constant
55 -1 (p-1)2 2 2.82641999706759157555 Murata's Constant
56 -1 (p+1)2 2 1.26655850147152857161  
57 -1 p(p2-1) 2 1.23129114888860356277  
60 -1 p2-p-1 2 2.67411272557002150896  
61 -1 p2+p-1 2 1.41956288050548591932 Inverse of 10 above.

References

Artin's Constant 0.37396

Density of the set of primes q, relative to the set of all primes, such that a given positive integer (not a proper power and with squarefree part incongruent 1 mod 4) is a primitive root modulo q. (more on FMC)

Rank 2 Artin Constant 0.69750

This, as well as the next two 0.85654 and 0.93127, are higher analogues of Artin's Constant, and are related to (among other things) the generation of prime residue class groups modulo a prime p by multiplicatively independent sets of r positive integers, a problem first studied by Keith Matthews. The case r=1 corresponds to Artin's Constant.

(The `other things' include generalizations of Artin's Conjecture to number fields of degree r, this being the context in which these constants recently starred in Hans Roskam's thesis.)

More on FMC and in the papers listed below.

Twin Prime Constant 0.66016

This is part of a conjectural density formula for the number of twin primes not exceeding a given bound. (more on FMC)

Hardy-Littlewood Constants C3=0.63517, C4=0.30749, C5=0.40987

Part of an infinite family of which the Twin Prime Constant is the first member; the general formula for Cn has

g-f=pn-1(p-n)

and

g=(p-1)n

and p0 the first prime larger than n.

For these and the derived constants D=(9/2)C3 and E=(27/2)C4 see again FMC.

"Carefree" Constant 0.42825 and "Strongly Carefree" Constant 0.28675

Let us call a pair (a,b) of natural numbers "carefree" if a is squarefree and coprime to b, "strongly carefree" if, in addition, b is also squarefree. The sets of such pairs have natural densities 0.42825 and 0.28675, respectively, relative to all pairs of positive integers.

Moreover, 0.28675 is also the natural density of pairwise coprime triples of positive integers (relative to all such triples).

See sections 2.7, 4.4, 4.7 in Schroeder's book. (more on FMC)

Quadratic class number Constant 0.88151

This is related to the average behavior of class numbers of real quadratic fields. See Cohen's book, section 5.10.1, page 290 or 291 (depending on the edition).

Feller-Tornier Constant 0.32263

The density of natural numbers whose prime factorization contains an even number of distinct primes to powers larger than the first (ignoring any prime factors which appear only to the first power) equals (1 + 0.32263...)/2=0.661317....

Note that up to a factor 1/zeta(2), this can equally well be computed from the entry number 15.

Sarnak's Constant 0.72365 or 0.74397

The second value, which is 5pi2/48 times the original product, is the one of interest in Sarnak's work about class numbers. (more on FMC)

Stephens' Constant 0.57596

Density (up to a rational factor) of the set T(a,b) of primes q such that q divides ak-b for some k, given multiplicatively independent integers a and b. (more on FMC)

Heath-Brown and Moroz's Constant 0.0013176

This is one factor in a formula for the number of primitive points of height not exceeding a given value on a cubic surface. Think of 1-f/g as coming from

prod (1 - 1/p)7 (1 + (7p+1)/p2).

(more on FMC)

Totient Constant 2.20386

Slightly easier to compute as zeta(2) times the value 1.33978 obtained as shown above, this equals the sum

sumn>0 1 / (n phi(n))

where phi denotes the Euler totient function. It appears in the paper by Stephens mentioned above. Pieter Moree has recently improved some of the estimates from that paper.

A related constant, which should be computable in a similar fashion from logarithms of zeta values, is the sum over the 1/phi(n) themselves, which was studied by Landau.

Murata's Constant 2.82642

This belongs into a context closely related to Artin's Constant. (more on FMC)


G. Niklasch / Fri, 23 Aug 2002 14:15:53 MEST