Beurling considered a sequence of reals $1<x_1<x_2<\cdots <$ as "primes" and then the ordered sequence of all products of these "primes" as "integers". Let us consider Beurling primes which are small perturbations of the ordinary primes. My question is:
1) How small perturbation guarantees that the Beurling primes/integers satisfy the prime number theorem.
2) How small perturbation guarantees that the system of Beurling primes/integers satisfies the RH given the "ordinary" RH.
3) How small perturbation guarantees that the the system of Beurling primes/integers satisfies Cramér's conjecture ("the number of integers between the $n$th prime ($p_n$) and the $n+1$ prime is $O(\log^2 p_n)$) given the "ordonary Cramér's conjecture".
Remark: I left the notion "small perturbation" vague and there is some other vagueness in the questions. But I will be interested in answers for any variant. Other basic results/conjectures can also be considered, and I will be interested also in information about them.
Motivation: This is motivated by some questions raised during the discussion of polymath 4 that I recalled in the context of the recent question The enigmatic complexity of number theory .
Clarification: (In response to Lucia.) I am looking for results of the following kind. (1) If $x_n=p_n+f(n)$ then the Beurling system based on $x_n$ satisfies the PNT. (2) If $x_n=p_n+g(n)$ then assuming RH, the Beurling system based on $x_n$ satisfies the assertion of the RH. (3)If $x_n=p_n+h(n)$ then assuming Cramér's conjecture, the Beurling system based on $x_n$ satisfies the assertion of Cramér conjecture.
