We introduce a sparse zeta-like function S (alpha) = sum from n=1 to infinity of Pₙ to the power -alpha, where Pₙ 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 alpha₀ approximately 1. 137839172734383739134422566646 (denoted Po₁) satisfying S (Po₁) = 1/zeta (2) = 6/pi². 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).
Robert Benjamin (Fri,) studied this question.