Equivalents of the Riemann Hypothesis

Abstract

Using a result of recursive function theory and results of the complex analysis of Takeuti which is based on a type theory and the work of Kreisel, and which gives a conservative extension of first order Peano arithmetic (PA), it is shown that the Riemann hypothesis is decidable in PA.

On the Riemann Hypothesis for the Zeta Function

“…Remark. It is known that ( 45) and ( 47) have been proved by Hadamard [8], (45) has been reported in [3], and ( 47 [46]. Now, we start with the work from Hadamard to Patterson.…”

“…Proof. See [18] and related references ([6], p.288; [46], p.49).…”

“…Remark. Here, ξ (0) = ξ (1) = 1/2 and ξ (1/2) ≃ 0.994 ( [46], p.49).…”

“…Theorem 4. If µ is defined in (46), then for s ∈ C and n ∈ N,…”

“…where µ is defined in (46) and the product runs over zeroes s n of (4). Combining (127) and Theorem 11, we find that there exist real zeros…”