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

- P. Moree,
Approximation of singular series and automata, to appear in
*Manuscripta Math.*(1999); Preprint available (ps).

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-`

with rational coefficients,
where the degree of the polynomial *f*/*g**g* is at least 2 plus
that of *f*,
evaluated at all primes

, or sometimes
at almost all primes (e.g. when one factor would vanish for
*p*

).*p*=2

Products of terms of the shape
`1+`

are readily accommodated by
moving the sign into *f*/*g*

. What really counts
for the computation, however, is the behavior of
*f*

and of *g*-*f*

.
*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

`prod`

,_{k>1}zeta(k)^{-e[k]}

however, although the convergence tends to be a lot better
due to the exponential convergence of `zeta(`

to 1 as *k*)

increases, it is still unsatisfactory,
and this product does not converge at all when the exponents
*k*

grow too fast.*e*[*k*]

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(`

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.
*k*

There are some hidden duplications among our constants, in the
following sense: One can introduce or remove factors of

in
*p*^{2}/(*p*^{2}-1)`(`

, and compensate for
this by writing a suitable power of *g*-*f*)/*g*`6/`

(inverse of *pi*^{2}`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

and *g*

swapping rôles.*g*-*f*

This table provides an overview.

Table entries are structured as follows: entry number;
numerator

, denominator *f*

and starting prime *g*

in the
original formula*p*_{0}

`prod`

,_{p}(1 -f(p)/g(p))`;`

p>=p_{0}

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

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 *f*

are of simple shape:
often the numerator is 1, and *g*

is a product of
factors *g*

and *p*`(`

, 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.*p*±1)

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

by multiplicatively independent
sets of *p*

positive integers, a problem first
studied by Keith Matthews. The case *r*

corresponds
to Artin's Constant.*r*=1

(The `other things' include generalizations of Artin's Conjecture
to number fields of degree

, this being the
context in which these constants recently starred in
Hans Roskam's
thesis.)*r*

More on FMC and in the papers listed below.

- K. R. Matthews,
A generalisation of Artin's conjecture for primitive roots,
*Acta Arith.*29 (1976) 113-146; MR 53 #313. - F. Pappalardi,
On the r-rank Artin conjecture,
*Math. Comp.*66 (1997), 853-868; MR 97f:11082. - L. Cangelmi and F. Pappalardi,
On the r-rank Artin conjecture II,
*J. Number Theory*75 (1999) 120-132. - H. Roskam, Artin's primitive root conjecture for quadratic fields, preprint available in PostScript. (1999).

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

*C*_{3}=

0.63517,
*C*_{4}=

0.30749,
*C*_{5}=

0.40987Part of an infinite family of which the Twin
Prime Constant is the first member; the general formula for

has*C*_{n}

g-f=p^{n-1}(p-n)

and

g=(p-1)^{n}

and

the first prime larger than
*p*_{0}

.*n*

For these and the derived constants

and
*D*=(9/2)*C*_{3}

see again
FMC.*E*=(27/2)*C*_{4}

Let us call a pair `(`

of natural
numbers "carefree" if *a*,*b*)

is squarefree
and coprime to *a*

, "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.*b*

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)

- M. R. Schroeder,
*Number Theory in Science and Communication: With Applications in Cryptography, Physics, Digital Information, Computing and Self-Similarity*, 2nd ed., Springer-Verlag 1986. (There is a 1997 edition, too) - P. Moree, Counting carefree couples, unpublished manuscript (1999); available from FMC as DVI file.

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).

- H. Cohen,
*A Course in Computational Algebraic Number Theory*, Springer-Verlag, 1993; MR 94i:11105.

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**.

- W. Feller and E. Tornier,
Mengentheoretische Untersuchungen von Eigenschaften
der Zahlenreihe,
*Math. Ann.*107 (1933), 188-232

The second value, which is `5`

times
the original product, is the one of interest in Sarnak's work about
class numbers.
(more on FMC)*pi*^{2}/48

- P. C. Sarnak,
Class numbers of indefinite binary quadratic forms II,
*J. Number Theory*21 (1985) 333-346; MR 87h:11027.

Density (up to a rational factor) of the set
`T(`

of primes *a*,*b*)*q*
such that *q* divides

for some *a*^{k}-*b**k*, given multiplicatively independent integers
*a* and *b*.
(more on FMC)

- P.J. Stephens,
Prime divisors of second-order linear recurrences, I,
*J. Number Theory*8 (1976) 313-332; MR 54 #5142. - P. Moree and P. Stevenhagen,
A two variable Artin conjecture,
*submitted*(1999); preprint available (DVI).

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-`

as coming from*f*/*g*

`prod (1 - 1/`

.p)^{7}(1 + (7p+1)/p^{2})

- D. R. Heath-Brown and B. Z. Moroz,
The density of rational points on the cubic surface
X
_{0}^{3}=X_{1}X_{2}X_{3},*Math. Proc. Cambridge Philos. Soc.*125 (1999) 385-395.

Slightly easier to compute as `zeta(2)`

times the
value 1.33978 obtained as shown above, this equals the sum

`sum`

_{n>0}1 / (nphi(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(`

themselves, which was studied
by Landau.*n*)

- E. Landau,
Über die zahlentheoretische Funktion phi(n) und ihre Beziehung
zum Goldbachschen Satz.
In:
*Collected works*Vol.1, 106-115.

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

- L. Murata,
On the magnitude of the least prime primitive root,
*J. Number Theory*37 (1991) 47-66; MR 91j:11082.

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