2011/03/14 by Michel Planat, M. Planat, Fabio Anselmi +3
Mathematics · Physics and Astronomy · #Advanced Operator Algebra Research #Analytic Number Theory Research #Context (archaeology) #Dedekind cut #Euler's formula #Function (biology) #Goldbach's conjecture #Limits and Structures in Graph Theory #Modulo #Pauli exclusion principle #Riemann hypothesis #Unitary state #math-ph #math.MP #quant-ph
paper · pdf · doi:10.1007/s11232-012-0074-x
published as Theoretical and Mathematical Physics 171, 3 (2012) 780-791 · 11 pages
arxiv created 2011/03/14 · openalex publication_date 2012/06/01 · arxiv updated 2015/05/27 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
Let consider the Pauli group Pq=<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> =ωs |s>, with ω=exp(2iπ/q). It has been found that the number of maximal mutually commuting sets within Pq is controlled by the Dedekind psi function ψ(q)=q ∏p|q(1+(1)/(p)) (with p a prime) \citePlanat2011 and that there exists a specific inequality (ψ(q))/(q)>eγlog log q, involving the Euler constant γ∼ 0.577, that is only satisfied at specific low dimensions q ∈ \mathcal 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 p2) is equivalent to Riemann hypothesis. Introducing the Hardy-Littlewood function R(q)=2 C2 ∏p|n(p-1)/(p-2) (with C2 ∼ 0.660 the twin prime constant), that is used for estimating the number g(q) ∼ R(q) (q)/(ln2 q) of Goldbach pairs, one shows that the new inequality (R(Nr))/(log log Nr) \gtrapprox eγ is also equivalent to Riemann hypothesis. In this paper, these number theoretical properties are discusssed in the context of the qudit commutation structure.