Five Compact Exponential Formulas for the Distribution of Primes

We collect five self-contained closed-form formulas that approximate with high accuracy: (1) the n-th prime pn; (2) the prime-counting function π(x); (3) the logarithm of the primorial pn# in terms of n; and (4)–(5) fast inverses linking pn and π(x). All coefficients depend only on elementary constants (π) and rational numbers. Numerical tests show sub-percent errors from n ≥ 100 or x ≥ 104 upward. No facts about the zeros of the Riemann zeta function are assumed.