Limit of Barrett’s prime-counting formula to the prime number theorem

Determine whether there exists a limiting process applied to the Barrett prime-counting series Barr(n)—defined as 3 plus the sum from k = 5 to n − 1 of sin(pi*(k − 1)!/k) divided by sin(pi/k)—that yields the prime number theorem asymptotic for the prime-counting function, namely pi(n) ~ n / ln(n), as n tends to infinity.

Background

The paper presents Rafael Barrett’s 1903 prime-counting construction based on Wilson’s theorem, yielding an exact counting series Barr(n) expressed via trigonometric terms that evaluate to 0 or 1 depending on whether k is composite or prime. The author then poses a concluding question about connecting this exact counting series to the asymptotic behavior known from the prime number theorem.

The open problem asks whether a suitable limit of Barrett's series can reproduce the classical asymptotic pi(n) ~ n / ln(n), thereby bridging Barrett’s exact discrete counter with the continuous asymptotic distribution established by Hadamard and de la Vallée Poussin.