Prime Residue Class of Uniform Charges on the Integers

• Published in: Journal of theoretical probability Vol. 33; no. 1; pp. 340 - 360
• Main Authors: Spece, Michael, Kadane, Joseph B.
• Format: Journal Article
• Language: English
• Published: New York Springer US 01-03-2020; Springer Nature B.V
• Subjects: Integers; Mathematics; Mathematics and Statistics; Prime numbers; Probability Theory and Stochastic Processes; Statistics; Prime numbers; Uniform distribution; Finite additivity; 60A05; Residue class; Probability charge
• ISSN: 0894-9840; 1572-9230
• DOI: 10.1007/s10959-018-0860-y

There is a probability charge on the power set of the integers that gives probability 1 / 
p
to every residue class modulo a prime
p
. There exists such a charge that gives probability
w
to the set of prime numbers iff
w
∈
[
0
,
1
/
2
]
. Similarly, there is such a charge that gives probability
x
to a residue class modulo
c
, where
c
is composite, iff
x
∈
[
0
,
1
/
y
]
, where
y
is the largest prime factor of
c
.