SWhat changed
Independent analytic proofs established the asymptotic density of primes, in the weighted form θ(x)=Σ_{p<x}log p∼x and the familiar equivalent count form π(x)∼x/log x. Hadamard's 1896 paper proves the absence of zeta zeros on Re(s)=1 away from the pole, then obtains the asymptotic by a convergent smoothed contour integral and removes the smoothing. The same paper treats fixed reduced arithmetic progressions, with weighted density1/φ(k); de la Vallée Poussin's own Congress report states the corresponding weighted and counting asymptotics.
Scientific scope
Independent analytic proofs established the asymptotic density of primes, in the weighted form θ(x)=Σ_{p<x}log p∼x and the familiar equivalent count form π(x)∼x/log x. Hadamard's 1896 paper proves the absence of zeta zeros on Re(s)=1 away from the pole, then obtains the asymptotic by a convergent smoothed contour integral and removes the smoothing. The same paper treats fixed reduced arithmetic progressions, with weighted density1/φ(k); de la Vallée Poussin's own Congress report states the corresponding weighted and counting asymptotics. This is an asymptotic ratio as x tends to infinity, not an exact finite prime count, a uniform explicit remainder, a prime-location algorithm or the Riemann hypothesis. For progression m mod k, k is fixed and gcd(m,k)=1; no uniformity in a growing modulus or nonreduced classes. The linked full proof is Hadamard's 1896 paper; de la Vallée Poussin's supplied source is his own later two-page congress report, rather than his 1896 memoir. The equivalent modern π notation is explanatory rather than a verbatim Hadamard formula. Quadratic-form extensions and later zero-free regions/elementary proofs remain outside the admitted claim.
Evidence
The linked original works support the scoped statement and contribution attribution. WRS has not independently replayed experiments or reconstructed a full mathematical proof. This is an asymptotic ratio as x tends to infinity, not an exact finite prime count, a uniform explicit remainder, a prime-location algorithm or the Riemann hypothesis. For progression m mod k, k is fixed and gcd(m,k)=1; no uniformity in a growing modulus or nonreduced classes. The linked full proof is Hadamard's 1896 paper; de la Vallée Poussin's supplied source is his own later two-page congress report, rather than his 1896 memoir. The equivalent modern π notation is explanatory rather than a verbatim Hadamard formula. Quadratic-form extensions and later zero-free regions/elementary proofs remain outside the admitted claim.
Contributors
- The named authors in the research record · SCIENCE: Independent Hadamard and de la Vallée Poussin contributions are one result family, not a joint paper byline. The source set distinguishes the original Hadamard proof from the other investigator’s later own report.
- Sur la distribution des zéros de la fonction ζ(s) et ses conséquences arithmétiques (1896 original)
- Sur la théorie des nombres premiers (own 1897 Congress report, published1898; printed194–195)
Provisional WRS assessment, edition 2026-10-23.3. Independent expert calibration is pending. This grade does not certify correctness or assign ownership.
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