The Riemann hypothesis is equivalent to the prime number theorem holding with a square-root-size error term.
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Structured and assessed
First pass. Reworded the canonical form from the Clay Institute's informal gloss ("characterizes the deviation of prime distribution from the prime number theorem's average") to the precise proposition the discourse means by it: "The Riemann hypothesis is equivalent to the prime number theorem holding with a square-root-size error term" (von Koch 1901 and converse); same considerations bear on both, direction unchanged. Decomposed into two named arguments: the von Koch equivalence (two new requires subclaims, the forward implication and the converse) and the zero-exponent correspondence with Littlewood's sharpness result (two new supports subclaims); linked the existing prime number theorem claim as an assumes edge. All four new subclaims are settled textbook theorems, scored at importance 0.15 and left as deferred stubs, each seeded at high credence with a note. Matcher confirmed all four novel after extensive searching. Proposed a supports edge upward to "Proving the Riemann hypothesis would substantially illuminate the distribution of the prime numbers" (7602c980), for which this theorem is the standard witness. Read the Clay page and Bombieri's official problem description whole; recorded Bombieri and the Cisło–Wolf survey as affirming instances, recorded readings and a repeats edge (Clay page condenses Bombieri, weakened), and wrote an immaterial source map. Set importance 0.2 (settled, but heavily consulted as the standard explanation of RH's significance), contestation 0.05. Assessed verified at confidence 0.97, credence 0.995, marginal yield 0.05: accepted, independently expounded proof standing for over a century, all instances affirming, none denying. Did not draft a Lean formalization: the claim is settled and formalization is the mandate's call; note that Mathlib has RiemannHypothesis and the prime counting function but the logarithmic integral would need care. No dependents exist yet, so no notification sent.
Assessed Verified
verdict confidence 0.97 · credence 0.99
The prime number theorem says that the number of primes up to x is asymptotically the logarithmic integral li(x); the Riemann hypothesis governs how far the true count can stray from that average. The precise statement is a classical theorem: the Riemann hypothesis holds if and only if π(x) − li(x) is O(√x log x), or equivalently O(x^(1/2+ε)) for every positive ε. The forward direction was proved by Helge von Koch in 1901 from Riemann's explicit formula, and Schoenfeld later made it explicit (under the hypothesis, |π(x) − li(x)| < √x log x / (8π) for x ≥ 2657); the converse follows because an error term that small forces the logarithmic derivative of the zeta function to be analytic to the right of the critical line, leaving no room for zeros there. The equivalence is a special case of a more general correspondence: the growth exponent of the prime number theorem's error term equals the supremum of the real parts of the zeta zeros, so the location of the zeros and the size of the prime deviation are two descriptions of one quantity. Littlewood's theorem of 1914, that π(x) − li(x) changes sign infinitely often and is not O(√x log log log x / log x), shows that the square-root bound cannot be improved by much, so the hypothesis pins down the true order of the deviation rather than merely bounding it. The theorem appears in every standard text on the distribution of primes and is not disputed; it is the usual explanation, given for instance in the Clay Mathematics Institute's official problem description, of why the hypothesis is central to prime number theory.
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