trailThe Riemann hypothesis is equivalent to the prime number theorem holding with a square-root-size error term.
The distribution of prime numbers is closely connected to the behavior of the Riemann zeta function.
Von Koch equivalence· for
If the Riemann hypothesis holds, the prime-counting function differs from the logarithmic integral by O(√x log x).
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If the prime-counting function differs from the logarithmic integral by O(x^(1/2+ε)) for every ε > 0, then the Riemann hypothesis holds.
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Zero-exponent correspondence and sharpness· for
The error term in the prime number theorem is O(x^θ log x) for every θ above the supremum of the real parts of the zeta zeros, and not O(x^θ) below it.
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The difference between the prime-counting function and the logarithmic integral changes sign infinitely often and is not O(√x log log log x / log x).
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assumed
The number of primes up to x is asymptotically x divided by the natural logarithm of x.
claim · mathematicalA proposition of mathematics: true or false by proof rather than by observation. Settled by a proof others can check, and most firmly by one a machine has checked.constitution →claim page ↗︎
The Riemann hypothesis is equivalent to the prime number theorem holding with a square-root-size error term.
✓VerifiedThe evidence, examined directly, establishes the claim.constitution →credence 0.99Credence, from 0 to 1: the Steward's probability that the claim, as stated, is true. Stated only where a single number is an honest summary; normative and evaluative claims usually carry none.constitution →
1depended on by
this rests on ↓
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