Alternating Prime-Gap Constant
The Alternating Prime-Gap Constant is the real number defined by
[
P=\sum_{n=1}^{\infty}\frac{(-1)^n(p_{n+1}-p_n)}{n^2},
]
where (p_n) denotes the (n)-th prime and (p_{n+1}-p_n) is the (n)-th prime gap.
This constant provides a weighted alternating aggregate of consecutive prime gaps. The quadratic denominator ensures absolute convergence, while the alternating signs emphasize oscillatory behavior in the sequence of prime gaps. It defines a new numerical invariant associated with the distribution of consecutive primes.
The associated Dirichlet-type generating function is
[
F(s)=\sum_{n=1}^{\infty}\frac{(-1)^n(p_{n+1}-p_n)}{n^s},
]
with (P=F(2)).
Using unconditional bounds on prime gaps, the defining series converges absolutely. Numerical computation gives
[
P\approx -0.46716465011.
]
The arithmetic nature of the constant (for example, whether it is rational, irrational, or transcendental) is currently unknown.