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 decreases, 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.
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 (p±1), 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.
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)
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.
This is part of a conjectural density formula for the number of twin primes not exceeding a given bound. (more on FMC)
C3=0.63517,
C4=0.30749,
C5=0.40987Part 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.
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)
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).
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.
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)
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)
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).
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.
This belongs into a context closely related to Artin's Constant. (more on FMC)