Version 3. We study the shifted spectral sequence associated with the nontrivial zeros of the Riemann zeta function and prove, as the main closed result of the paper, that the Riemann Hypothesis is equivalent to this sequence being a Hausdorff moment sequence. Under RH, the representing measure is explicit. The converse is obtained through a non-cancellation argument: any zero off the critical line produces a genuine pole of the generating function at z = rho(1 - rho), while the Hausdorff representation extends the same function holomorphically to the slit plane C \ [1, infinity). The paper also develops a wider positivity framework around the cumulant power sums of the Riemann xi function, including complete monotonicity, shifted Hankel positivity, Stieltjes structure, Pick-type consequences, and Cauchy-Binet positivity. This version presents the main equivalence theorem in a more consolidated, direct, and self-contained form.
armando luis francisco (Sun,) studied this question.