From the Encoding of Prime Gaps by SeqImpar and SeqPar to the Correspondence Between Zeros and Poles of the Riemann Transform

This article organizes and proves the mathematical chain connecting the sequencesSeqImpar and SeqPar to the gaps between consecutive odd primes and, through a Stieltjes–Laplace transform, to the nontrivial zeros of the Riemann zeta function. First, closed formsfor the sequences are derived, and it is shown that the ratio of their increments makes itpossible to recover the next prime exactly when the preceding prime and the observedratio are known. Next, the discrepancy function for prime counting and its transform arereconstructed. The prime zeta function is then related to ζ(s) through the Euler product;an inversion obtained from a divisor recurrence recovers the coefficients µ(m)/m. Thecorrection 12Li(√x) is characterized by the removal of a spurious pole of the transform.Finally, it is shown that each nontrivial zero ρ with ℜρ > 1/2 corresponds to a pole of L(z)at z = ρ−1/2, whose residue is the negative of its multiplicity. This yields the equivalenceRH ⇐⇒ L has no poles in ℜz > 0. The equivalence is a rigorous reformulation of theRiemann hypothesis, not a proof of it.