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    "title": "Hadamard and de la Vallée Poussin: the prime number theorem",
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      "statement": "# Hadamard and de la Vallée Poussin: the prime number theorem\n\n## Prime numbers thin out, but how predictably?\n\nA prime number has exactly two positive divisors, one and itself. As numbers grow, primes become less frequent, yet they do not follow a simple repeating spacing pattern. The prime number theorem gives a precise limiting rule for their cumulative count. If pi(x) denotes the number of primes up to x, then its ratio to x divided by the natural logarithm of x tends to one as x grows without bound.\n\n## Check a small table\n\nThere are 25 primes up to 100, while 100 divided by its natural logarithm is about 21.7. There are 168 up to 1,000, compared with an estimate of about 144.8. These elementary classroom checks illustrate a rough prediction, not impressive precision at small inputs. Use the natural logarithm, often written ln, rather than the base-ten logarithm. Substituting another logarithm changes the constant and therefore the theorem's statement.\n\n## A ratio approaching one is not an error approaching zero\n\nSuppose a sequence of estimates and true counts have a ratio tending to one. Their absolute difference can still increase. For a simple invented comparison, n squared and n squared plus n have ratio tending to one even though their difference is n. The prime number theorem is likewise a relative asymptotic statement. It is not an exact finite prime-count formula, a bound on every gap or a fast algorithm that locates the next prime.\n\n## Two independent investigators, one result family\n\nHadamard and de la Vallee Poussin independently obtained analytic proofs. Their contributions are not a joint paper, and the catalogue does not count the same prime-density law twice. The linked source set includes Hadamard's full 1896 paper and the other investigator's own later congress report. Work on a zeta-function zero boundary enters the proof, but this is not a proof of the Riemann hypothesis. Fixed arithmetic progressions also require their stated conditions.\n\n## A small agent project\n\nAsk your agent to produce and verify a prime-count table for a bounded range, recording the counting convention and logarithm used. Include the true count, the estimate and their ratio. Have it explain why a few successful rows cannot establish the infinite limiting theorem. A useful follow-up could trace the exact analytic statement in the original paper or distinguish the previously known prime-count bounds from the new asymptotic, rather than presenting a plot as a proof.\n\n## Scientific scope\n\nThis 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.\n\n## Sources for this explanation\n\n- [Sur la distribution des zéros de la fonction ζ(s) et ses conséquences arithmétiques (1896 original)](https://www.numdam.org/item/10.24033/bsmf.545.pdf): Pinned original contribution and stated hypotheses or experimental limits.\n- [Sur la théorie des nombres premiers (own 1897 Congress report, published1898; printed194–195)](https://www.e-rara.ch/download/pdf/2522595.pdf): Pinned original contribution and stated hypotheses or experimental limits.\n",
      "summary": "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.",
      "title": "Hadamard and de la Vallée Poussin: the prime number theorem"
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      {
        "title": "Prime numbers thin out, but how predictably?",
        "text": "A prime number has exactly two positive divisors, one and itself. As numbers grow, primes become less frequent, yet they do not follow a simple repeating spacing pattern. The prime number theorem gives a precise limiting rule for their cumulative count. If pi(x) denotes the number of primes up to x, then its ratio to x divided by the natural logarithm of x tends to one as x grows without bound."
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        "title": "Check a small table",
        "text": "There are 25 primes up to 100, while 100 divided by its natural logarithm is about 21.7. There are 168 up to 1,000, compared with an estimate of about 144.8. These elementary classroom checks illustrate a rough prediction, not impressive precision at small inputs. Use the natural logarithm, often written ln, rather than the base-ten logarithm. Substituting another logarithm changes the constant and therefore the theorem's statement."
      },
      {
        "title": "A ratio approaching one is not an error approaching zero",
        "text": "Suppose a sequence of estimates and true counts have a ratio tending to one. Their absolute difference can still increase. For a simple invented comparison, n squared and n squared plus n have ratio tending to one even though their difference is n. The prime number theorem is likewise a relative asymptotic statement. It is not an exact finite prime-count formula, a bound on every gap or a fast algorithm that locates the next prime."
      },
      {
        "title": "Two independent investigators, one result family",
        "text": "Hadamard and de la Vallee Poussin independently obtained analytic proofs. Their contributions are not a joint paper, and the catalogue does not count the same prime-density law twice. The linked source set includes Hadamard's full 1896 paper and the other investigator's own later congress report. Work on a zeta-function zero boundary enters the proof, but this is not a proof of the Riemann hypothesis. Fixed arithmetic progressions also require their stated conditions."
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