argumentA line of reasoning from premises to a conclusion. Like a claim, it belongs to no one who makes it.constitution →

Zero-exponent correspondence and sharpness

forThis argument, if it holds, bears in favour of the claim.constitution →The Riemann hypothesis is equivalent to the prime number theorem holding with a square-root-size error term.

The argument

Given that the growth exponent of the prime number theorem's error term equals the supremum of the real parts of the zeta zeros, the Riemann hypothesis (supremum one half) is exactly the statement that this exponent is one half; and because the deviation is unconditionally at least of order √x up to logarithmic factors, the square-root bound is essentially the smallest the deviation could have, so the hypothesis pins down the true size of the deviation rather than merely bounding it.

Does it work?

Not yet evaluated.

Premises

UnassessedNo current assessment. Attention goes where its expected value is highest and someone funds it; nothing has funded an assessment of this claim yet, and anyone can.constitution →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.
UnassessedNo current assessment. Attention goes where its expected value is highest and someone funds it; nothing has funded an assessment of this claim yet, and anyone can.constitution →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).

Where it is made

Passages where sources make, criticize or mention this argument. They record where it appears, not how strong it is, and none has precedence over another.

No source is recorded as making it.

Earlier ratings

Sep 17, 2026
The inference goes through: if the error exponent equals the supremum of the zeros' real parts, then that supremum being one half is the same statement as the error term having exponent one half, and Littlewood's lower bound shows the resulting square-root order is the true order of the deviation, not merely an upper bound. The argument lives on the exponent correspondence, a standard theorem in Ingham's tract, with Littlewood's oscillation theorem supplying the sharpness; both are settled and the argument corroborates the direct equivalence rather than being needed for it.