Hi,
Short answer :
1.Yuri Matiysevich's undecidability (1970) in his MRDP theorem and his statement on the unsolvability of Hilbert 10th problem, adapted in his own investigations of Riemann problem.
2.Turing definition of uncomputability in his study of Riemann problem 1937 - 1953( Proc London Math Soc,3:99-117,1953 ).
3.Unpredictability in number - theoretical sense by G.Hardy - Montgomery - Keating (2020), and Tao's theory of " quasi-randomness of primes".
Long answer:
FORGOTTEN TURING MEANINGS.
Alan Turing exploits in code theory,AI and foundations of computer sciences are well-known for many. Less well known is that Turing was also interested in the search of alternative and counterexample for Riemann Hypothesis (RH).These interests culminated in 2 computer programs that he implemented on the " Manchester Mark 1"(MM1) in 1949 - 1950. Turing's 2 programs on MM1 fall squarely within Euclid - Eratosphenes styles of thinking on primes. Turing developed UUU adaptations for RH and he worked on the improvement to Skews's theorem (1933) hoping to remove RH. He had hoped that the MM1 would find a counterexample for Riemann Hypothesis. My essay demonstrates similar trend by justification of an existence of Quantum or Nonclassical number theory.
