A Primorial Sparse Zeta Function with Root S(α) = 1/ζ(2)

We introduce a sparse zeta-like function S(alpha) = sum from n=1 to infinity of P_n to the power -alpha, where P_n is the n-th primorial. The series converges absolutely for alpha > 0 and extends analytically to Re(alpha) > 0. It is continuous and strictly decreasing on (0, infinity), with limit as alpha -> 0+ of S(alpha) = +infinity and limit as alpha -> infinity of S(alpha) = 0. By the intermediate value theorem and strict monotonicity, there exists a unique positive real root alpha_0 approximately 1.137839172734383739134422566646 (denoted Po_1) satisfying S(Po_1) = 1/zeta(2) = 6/pi^2. High-precision numerical evaluation (mpmath, 80 primorial terms) confirms the error is less than 10 to the power of minus 40. The super-exponential growth of primorials ensures ultra-rapid convergence of partial sums; explicit tail bounds are derived using the prime number theorem. This ultra-sparse sum offers a novel computational link to the classical Basel constant zeta(2).