TITLE:
A Sigma-Based Detection Framework for Prime Constellations with Applications to Bounded Prime Gaps
AUTHORS:
Michael M. Anthony
KEYWORDS:
Twin Primes, Prime Constellation in p, p + 2N, Gauss Multiplication Formula, Gamma Function, Sum-of-Divisors, Sieve Theory, Zhang, Bounded Primes, Prime Detecstor Function, Dirichlet Series, Bombieri-Vinogradov, Hardy-Littlewood Conjecture
JOURNAL NAME:
Advances in Pure Mathematics,
Vol.16 No.7,
July
30,
2026
ABSTRACT: A family of prime constellation detectors
D(
p,N
)
expressed as rational powers of 2π, built from the sum-of-divisors function σ. The central object
D(
p,N
)=
(
2π
)
p+N+1−
[
σ(
p
)+σ(
p+2N
) ]/2
equals 1 if and only if both
p
and
p+2N
are prime, and is strictly less than 1 otherwise. This construction generalizes naturally to k-tuples via a product formula. A zeta-regularized product identity
∏
D(
p
)
=
(
2π
)
3
8
is created and decomposed into an associated sum of three explicitly characterized sub-series whose regularized total equals −5/48, a value determined purely by
ζ(
0
)
and
ζ(
−1
)
. It is shown that the composite sub-series
F(
1
)
converges to a constant numerically close to
log
10
2
. Zhang’s bounded-gap theorem is recast in this form, showing that the average sieve sum
T(
x,M
)>1
for
M≥6
, and extend the framework to k-prime clusters. The triple prime case D3(p) = 1 is proven to have exactly one solution (p = 3) via a complete mod-3 obstruction, validating the framework in a case where the answer is known.