  In this article, we explore the Riemann zeta function with a perspective on
primes and non-trivial zeros. We develop the Golomb's recurrence formula for
the $n$th+1 prime, and assuming (RH), we propose an analytical recurrence
formula for the $n$th+1 non-trivial zero of the Riemann zeta function. Thus all
non-trivial zeros up the $n$th order must be known to generate the $n$th+1
non-trivial zero. We also explore a variation of the recurrence formulas for
primes based on the prime zeta function, which will be a basis for the
development of the recurrence formulas for non-trivial zeros based on the
secondary zeta function. In the last part, we review the presented formulas and
outline the duality between primes and non-trivial zeros. The proposed formula
implies that all primes can be converted into an individual non-trivial zero
(assuming RH), and conversely, all non-trivial zeros can be converted into an
individual prime (not assuming RH). Also, throughout this article, we summarize
numerical computation and verify the presented results to high precision.
