Where did Riemann go wrong?

Where did Riemann go wrong? Why couldn’t he answer his own question?

This project does not claim that Riemann’s formulas are incorrect. The issue is not with the mathematics, but with the vantage point. Riemann succeeded in encoding the entire structure of the primes into the zeta function ζ(s) and its zeros in the complex plane. What he almost never touched was the layer underneath ζ(s): the raw 0/1 stream of primes, the δ-impulses on the natural line, the way they clump and thin out inside finite intervals.

His method was strictly “top–down”: start from ζ(s), study its zeros, and only then try to read off the behaviour of primes. In that picture he was looking at the shards of the vase after it shattered—the spectral trace in ζ(s)—instead of watching how the vase is built and how it cracks in real time. So he could formulate the hypothesis (all non-trivial zeros lie on Re(s) = 1/2), but not explain in everyday language why prime chaos should organise itself so that this line appears at all.

This work goes the opposite way—“bottom–up”.

1. It first introduces an FRA-language for the raw δ-history of primes: windows [a,b]; δ(n) as a “prime / no prime” impulse; a global form F2(x) as a smooth expectation; the difference field Φ and relative deviation ε as local chaos; corridors ∴, the background zone Ξ, rare deep holes and clusters, and Ø-windows where the usual smooth picture breaks.

2. Then this language is translated back into standard mathematics: precise definitions of δ(n), π(x), fixed-length windows, expected prime count F2(W), local error Φ(W), relative deviation ε(W), and statistics of prime gaps inside a window (F3-patterns A/B/C/D).

3. On this basis FRA-conjectures are stated that rephrase Riemann’s problem: not “where are the zeros of ζ(s)?”, but “how are the ε(W) chaos corridors organised, how rare are extreme windows and Ø-zones, and how much of the δ-history is in fact captured by F2 and a small set of typical local patterns?”.

The work also includes a concrete numerical experiment on [1…1 000 000]: real windows of length 100, 1 000 and 10 000; computed ε(W); a map of deep holes (“almost-Ø” windows); and a check that the FRA picture — background + rare strong anomalies + shrinking chaos corridors — matches the actual distribution of primes rather than living only as a philosophical story