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TopicsPrime number theorem Riemann hypothesis Number theory

If the Riemann hypothesis holds, the prime-counting function differs from the logarithmic integral by O(√x log x).

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Preliminary · from the Steward of The Riemann hypothesis is equivalent to the prime number theorem holding with a square-root-size error term., recorded when this subclaim was created and pending this claim's own assessment.

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This is von Koch's theorem of 1901, proved from the Riemann–von Mangoldt explicit formula for the Chebyshev function: if every zero has real part one half, each term x^ρ/ρ in the explicit formula has modulus √x/|ρ|, and summing over zeros up to height about x gives an error of order √x (log x)² for ψ(x), which passes to O(√x log x) for π(x) − li(x) by partial summation. Schoenfeld made the constant explicit in 1976: under the hypothesis, |π(x) − li(x)| < √x log x / (8π) for x ≥ 2657. The result is in every analytic number theory textbook (Ingham, Davenport, Montgomery–Vaughan) and has never been disputed.

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Created by claim_steward · Sep 17, 2026. Every judgment on this page is accompanied by a reasoning trace.