Pauli graphs, Riemann hypothesis, and Goldbach pairs

Abstract

Abstract.Let consider the Pauli group P q = X, Z with unitary quantum generators X (shift) and Z (clock) acting on the vectors of the q-dimensional Hilbert space via X |s = |s + 1 and Z |s = ω s |s , with ω = exp(2iπ/q). It has been found that the number of maximal mutually commuting sets within P q is controlled by the Dedekind psi function ψ(q) = q p|q (1 + 1 p ) (with p a prime) [2] and that there exists a specific inequality> e γ log log q, involving the Euler constant γ ∼ 0.577, that is only satisfied at specific low dimensions q ∈ A = {2, 3, 4, 5, 6, 8, 10, 12, 18, 30}. The set A is closely related to the set A ∪ {1, 24} of integers that are totally Goldbach, i.e. that consist of all primes p < n − 1 with p not dividing n and such that n − p is prime [5]. In the extreme high dimensional case, at primorial numbers N r , it is known that the inequality ψ(Nr) Nr log log Nr e γ ζ(2) (for every r > 2) is equivalent to Riemann hypothesis. Introducing the Hardy-Littlewood function R(q) = 2C 2 p|n p−1 p−2 (with C 2 ∼ 0.660 the twin prime constant), that is used for estimating the number g(q) ∼ R(q) q ln 2 q of Goldbach pairs, one shows that the new inequality R(Nr) log log Nr e γ is also equivalent to Riemann hypothesis. In this paper, these number theoretical properties are discusssed in the context of the qudit commutation structure.