SeqImpar and SeqPar: Analytic Characterizations of the Zeros of the Riemann Zeta and Dirichlet Beta Functions and a Criterion Equivalent to the Riemann Hypothesis

We study two rational sequences, called SeqImpar and SeqPar, originally defined byrecurrences and converging to the common limit 5. The ratio of their convergence errors tendsto 1/2. Using this ratio, we construct two Dirichlet series weighted by the von Mangoldtfunction and associated with the Riemann zeta function and the Dirichlet beta function. Weprove that these series admit meromorphic continuations whose poles in the critical stripcorrespond exactly to the zeros of the respective functions, with residues equal to minusone-half of their multiplicities. We then prove that an estimate for the partial sum weightedby SeqImpar and SeqPar is equivalent to the Riemann Hypothesis, and state a parallelcriterion for the Dirichlet beta function. The paper rigorously distinguishes proved identities,equivalences, and open problems: the coincidence of the number 1/2 with the critical linedoes not, by itself, constitute a proof of the Riemann Hypothesis.