Stephens' constant

In number theory, Stephens' constant expresses the density of certain subsets of the prime numbers. Let

a

{\displaystyle a}

and

b

{\displaystyle b}

be two multiplicatively independent integers, that is,

a

m

b

n

≠

1

{\displaystyle a^{m}b^{n}\neq 1}

except when both

m

{\displaystyle m}

and

n

{\displaystyle n}

equal zero. Consider the set

T

(

a

,

b

)

{\displaystyle T(a,b)}

of prime numbers

p

{\displaystyle p}

such that

p

{\displaystyle p}

evenly divides

a

k

−

b

{\displaystyle a^{k}-b}

for some power

k

{\displaystyle k}

. Assuming the generalized Riemann hypothesis, the density of the set

T

(

a

,

b

)

{\displaystyle T(a,b)}

relative to the set of all primes is a rational multiple of

C

S

=

∏

p

(

1

−

p

p

3

−

1

)

=

0.57595996889294543964316337549249669

…

{\displaystyle C_{S}=\prod _{p}{\bigg (}1-{\frac {p}{p^{3}-1}}{\bigg )}=0.57595996889294543964316337549249669\ldots }

(sequence A065478 in the OEIS)

Stephens' constant is closely related to the Artin constant

C

A

{\displaystyle C_{A}}

that arises in the study of primitive roots:

C

S

=

∏

p

(

C

A

+

1

−

p

2

p

2

(

p

−

1

)

)

(

p

p

+

1

+

1

/

p

)

.