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bibkey: baezduarte2003criterion authors: “Luis Báez-Duarte” year: 2003 title: “A new necessary and sufficient condition for the Riemann hypothesis” doi: null url: “https://arxiv.org/abs/math/0307215v1” claim: “The original coefficient normalization, Newton continuation and compact uniformity of Proposition 2.1, including the analytic value at one, yield the sufficient RH direction; the corrected signed Abel and beta estimates transfer unweighted Mertens power bounds to original coefficient decay.” strata_touched:

  • D5/S3/Analytic/SeriesInequalities/NormalizedPochhammerBounds
  • D5/S3/Analytic/SeriesInequalities/BaezDuarteNewtonMajorant
  • D5/S3/Analytic/SeriesInequalities/BaezDuarteQBounds
  • D5/S3/Weil/Analytic/BaezDuarteNewton
  • D5/S3/Weil/Analytic/BaezDuarteContinuation
  • D5/S3/Weil/Analytic/BaezDuarteMertensKernel
  • D5/S3/Weil/Analytic/BaezDuarteMertensTransfer license: “No license assertion is made; this note records mathematical facts and bibliographic metadata only.” triage: anchor

Normalized products and compact convergence

Verified locator

  • URL: https://arxiv.org/abs/math/0307215v1

https://arxiv.org/abs/math/0307215v1 is the retained primary version. The registered PDF extraction locates equation (1.5), Lemma 1.1, and Proposition 2.1 on PDF page 2, printed page 2. This worker uses that recorded source evidence; it does not claim a new online metadata verification.

Source scope and local refinements

The retained primary record is version 1, published and updated 2003-07-16T03:11:19Z. The record supplies neither a journal reference nor a DOI. The retained six-page PDF has SHA256 3d0bd39e223561516a617b437274409112ff31b5488b8214fff749c3e00e8d34.

On printed page 2, equation (1.5) defines the product of the factors 1 - s/r for r = 1,...,k. Equation (1.6) identifies its binomial notation. Lemma 1.1 states that each open disk admits a positive unspecified constant bounding the product by that constant times k^(-Re(s)).

The local producer is a repo-derived quantitative refinement: for every nonnegative radius R, every k at least one, and every complex z with norm at most R, it uses the explicit constant exp(R + R^2). Neither this explicit constant nor this closed-disk formulation is printed in Lemma 1.1. The normalization and zero-index companions are repo-derived algebraic bridges using pinned mathlib’s descending Pochhammer polynomial. They do not prove a full generalized complex-binomial bridge.

Proposition 2.1, also on printed page 2, assumes every-positive-epsilon decay with exponent -3/4 + epsilon/2, identifies the series with reciprocal zeta on the open half-plane Re(s) > 1/2, and states uniform convergence on its compact subsets. The local consumer is a repo-derived proof refinement of that compact-convergence argument: under the stated every-epsilon decay with exponent -3/4 + epsilon, it constructs an all-index nonnegative summable majorant for the actual repository coefficients. Rescaling epsilon relates the two hypotheses, but that source correspondence does not assert the full proposition. No reciprocal-zeta identity or RH direction is proved here.

The zero index and every finite-prefix term are retained. No nonpole, nonvanishing, real-only or nonempty-compact hypothesis is imposed. The printed positive 3/4 on page 5 and the Lemma 2.2 cross-reference are not silently corrected by this note.

Source standing: user-requested formalization of known literature, with the above repo-derived refinements. The retained public search is bounded; later and unread literature remains unknown. Oracle access was unavailable, so no Oracle validation, model diversity, exhaustive search, or newest-literature certainty is claimed. TauCeti and Formalpedia are search-only sources.

Original Q and unconditional coefficient bound

Lemma 2.1 on printed page 3 defines the unsigned series in (2.2), starting at the positive integer one, and states the half-power order bound (2.3). Remark 1.1 on page 1 states unconditional half-power decay of the actual coefficients. Equation (2.7) on page 4 gives their signed Mobius series; equation (2.8) and the subsequent double-series discussion use the unsigned series to justify a Newton interchange. The latter discussion cites Lemma 2.2 as printed; the Q estimate is actually labeled Lemma 2.1. This distinction is retained, not silently repaired. The PDF bytes above were reopened for this Q implementation, and printed pages 1, 3, 4 and 5 were inspected directly.

BaezDuarteQBounds uses the exact unsigned series with index n shifted to n+1. Its explicit bound 3/sqrt(k+1), for every natural k including zero, is a repo-derived quantitative refinement, not the literal source’s unspecified constant. The finite geometric estimate on the closed unit interval, and the finite split N/(k+1)+1/N for every positive natural N and every endpoint M, are repo-derived proof refinements. The split includes M<N and retains every original summand. Its square-root cutoff gives the all-index bound; thin companions give 3*k^(-1/2) for k>=1 and Q(0)<=2. The actual frozen coefficient owner and its existing Mobius HasSum give |c(k)|<=Q(k), hence unconditional coefficient bounds. No second arithmetic definition is introduced.

The source uses Euler-Maclaurin and Gamma asymptotics; those arguments and constants are not claimed to have been transcribed by the finite-split proof. Neither an original Newton HasSum/interchange/Dirichlet identification nor holomorphic extension nor either RH direction is established by this module. The reviewed P and compact-majorant content above remains unchanged in scope.

Newton identity on the initial half-plane

The double-series argument around equation (2.8), printed page 4 of the same https://arxiv.org/abs/math/0307215v1 source, identifies the Newton series with reciprocal zeta initially for Re(s)>1. BaezDuarteNewton now proves this precise initial-domain HasSum for the existing actual coefficients and normalized Pochhammer polynomials evaluated at s/2. This new owner does not change the historical scope statements about the three earlier modules above.

The proof is a repo-derived implementation of that identification. It combines the frozen Q and P estimates into a summable unsigned coupled kernel, with strict outer exponent -(Re(s)+1)/2 below -1. Eventual comparison keeps the entire finite prefix. The absolute Moebius bound supplies the signed kernel’s absolute summability; this is derived from Re(s)>1, not imposed as a premise. The frozen signed coefficient HasSum and pinned mathlib’s complex binomial series evaluate the two families of fibers. Positive-real-base complex-power identities and the existing Moebius Dirichlet product identify the sum with 1/riemannZeta(s). The zero arithmetic row is one, P(0,z)=1, P(1,z)=1-z, and s=2 leaves only the original coefficient c(0).

The full Proposition 2.1 continuation to Re(s)>1/2 under every-epsilon decay, holomorphic extension, and both RH directions remain outside this theorem. The source’s printed positive 3/4 and Lemma 2.2 reference caveats remain as recorded above. No new primary-source access, successful Oracle participation, or exhaustive literature search is claimed by this implementation.

Continuation under the original decay assumption

BaezDuarteContinuation proves the continuation and compact uniformity of Proposition 2.1 under every-positive-epsilon coefficient decay, together with the sufficiency direction of Theorem 1.1. The finite complex coefficients are bound to the existing real coefficients using even-zeta realness. Rescaling epsilon explicitly binds the two printed decay conventions.

The holomorphic sum is identified with (s-1)/riemannZeta₁(s) on the whole open half-plane Re(s)>1/2. At one it is zero. Off one it agrees with the raw reciprocal zeta. All compact subsets, including those containing one, have uniform convergence of the actual range partial sums. Analytic uniqueness applied to the entire multiplier supplies the product identity and excludes zeros in the right half of the critical strip; the existing reduction then yields standard RH. These are literature-attested statements with a repo-derived implementation through public mathlib analytic continuation APIs.

The six original paper pages were read for this continuation. Theorem 1.1 and Proposition 2.1 print the negative exponent; the page 5 proof paragraph prints positive 3/4. The negative exponent is used in the proof. Page 5 also prints a positive Abel-integral sign and a shifted transformed kernel; the actual coefficient requires the negative derivative integral at its actual index. The square substitution in the beta integral requires a factor 1/2. These are recorded mathematical corrections, not an author-issued erratum. No unproved printed identity is imported as a hypothesis. The Abel and beta transfer and the RH necessity direction are not claimed by this continuation module. Earlier module scope statements above remain historical and exact.

Signed Mertens transfer at the original coefficient index

BaezDuarteMertensKernel and BaezDuarteMertensTransfer prove the corrected coefficient-specific Abel and beta estimates used on printed pages 5 and 6. The actual kernel is f_k(x)=x^(-2)*(1-x^(-2))^k for x>=1. Its derivative is -2*x^(-3)*(1-x^(-2))^k+2*k*x^(-5)*(1-x^(-2))^(k-1), for k>=1. For the signed unweighted sum M(x) over 1<=n<=floor(x), the proven identity is c_k=-integral_(1,infinity) M(x)*f_k'(x) dx. The boundary term vanishes, and local and improper integrability are proved explicitly. In particular, mu(0)=0 removes only the zero term; mu(1)=1 remains present.

For real 0<=a<2, an eventual |M(x)|<=A*x^a bound is extended to x>=1 by the actual finite estimate |M(x)|<=floor(x)<=x. Put b=1-a/2>0. The two weighted derivative integrals give A*(Re(B(b,k+1))+k*Re(B(b+1,k)))=A*(1+b)*Re(B(b,k+1)). The inverse-square change of variables is proved with its Jacobian, including the factor one half in each kernel integral. The public complex beta recurrence, GammaSeq identity and GammaSeq convergence give a positive constant independent of k. Thus for all k>=1 the original coefficient satisfies |c_k|<=C*k^(a/2-1). Both a global and an eventual signed Mertens hypothesis are supported, with exactly the same original arithmetic sum.

These are repo-derived proof implementations and explicit corrections of the paper’s printed signs, indices and half factor, with no author-issued erratum claim. The RH-to-unweighted-Mertens prerequisite and the complete two-direction RH iff are not proved by these modules. The completed continuation sufficient direction also consumes the original complex coefficient cast binding directly.

The Scribe attribution distinguishes the source’s sufficient RH direction and Newton continuation from repo-derived representation bindings, the explicit regularized multiplier formulation, the corrected kernel identities, and the quantitative transfer over all 0<=a<2. The latter carry repo-derived provenance with this same typed Library acknowledgement; they are not claimed to be literal separately numbered statements in the paper.