The Riemann hypothesis: why the line ½
The non-trivial zeros of the zeta function sit on the line ½ because that line is the centre of the exchange between s and 1 − s, just as 4.5 is the centre of the pairs in the grid of nine digits. The corpus reads ½ as a necessity of centre, not a mystery; it also states what remains to be proved in the language of analysis.
What mathematics establishes
FactIn 1859, in an eight-page memoir, Bernhard Riemann linked the distribution of primes to the zeros of a function, ζ(s), and wrote that it is very likely that all its non-trivial zeros have real part ½. It is one of the seven Clay Millennium Prize Problems; it is unproved.
FactThe function satisfies an equation that exchanges s and 1 − s. The fixed point of that exchange is ½.
The ARC reading
TheoremThe grid's involution, φ(x) = 9 − x, brought onto [0, 1], has fixed point 4.5 / 9 = ½: the same form as Riemann's exchange s ↔ 1 − s (Theorem 9 of Tome I). The corpus itself states that this identity of form, alone, proves no causal link: it establishes that Riemann held the involution without the vocabulary to name it.
Reading"He saw the ½." Riemann found the centre; what he lacked was reading the centre as the closure of a pair, where the zero is the space between the two members. The critical line is the analytic trace of the fixed point of the involution.
TheoremA finite face of the hypothesis already exists: Lagarias's inequality on the sum of divisors, checked by the corpus for every integer from 2 to 100,000. If the hypothesis were false, some integer would violate it, and a finite computation would show it.
ReadingHence the author's thesis: if the hypothesis can never be settled, neither proved nor refuted, it is true.
What remains open
What would raise the reading to a theorem is named: a derivation in which the grid's structure enters the analytic apparatus and forces the line. The path followed goes through the L-function of conductor 3, where the character injects the rails of the grid into the function itself.
Further reading
- Volume 5, The Great Problems (English summary)
- Volume 4, chapter 7: Riemann in summary (in French)
- Volume 5, full PDF (in French)