Von Koch equivalence
The argument
Because the Riemann hypothesis yields the error bound O(√x log x) for the prime-counting function and, conversely, an error bound O(x^(1/2+ε)) for every ε forces every zero onto the critical line, the hypothesis and the square-root-size error term each imply the other, which is the claimed equivalence.
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 →If the Riemann hypothesis holds, the prime-counting function differs from the logarithmic integral by O(√x log x).
- 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 →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.
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 is immediate: two implications in opposite directions make an equivalence. Its weight rests entirely on von Koch's forward direction and the converse from the error bound to the hypothesis, both textbook theorems proved over a century ago and never questioned, so the argument establishes the claim outright.