I don't know if I have misunderstood the information from the section about the heuristic related to Cramér's conjecture and the known facts about Maier's theorem, from the Wikipedia [*Cramér's conjecture*](https://en.wikipedia.org/wiki/Cram%C3%A9r%27s_conjecture#Heuristic_justification). I would like to know if there is a clear criterion or rule about when one can use the heuristic given by Cramér random model for prime numbers.



> 
> **Question (Emphasized after I've read the comments).** When does a professional number theorist know that can to use Cramer's heuristics, to make a reasoning in his/her work? I mean how to know if in certain situations the heuristic that provide Carmér random model will provide good reasonings about the distribution of primes, versus situations in which this heuristic should be wrong. **Many thanks.**
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If there is a clear discussion about it in the literature feel free to answer as a reference request, and I try to search and read it from the literature. In other case feel free to illustrate different scenarios (I say examples) where it is appropriate or not to use Cramér heuristics in calculations in analytitc number theory.


For the statement of Maier's theorem see [1] or the corresponding Wikipedia.


References:
-----------


[1] Helmut Maier, *Primes in short intervals*, The Michigan Mathematical Journal, 32 (2): pp. 221–225 (1985).



Is there a clear criterion or rule about when one can use the heuristic given by Cramér random model for prime numbers?
Terry Tao's blog has the following [post](https://terrytao.wordpress.com/2015/01/04/254a-supplement-4-probabilistic-models-and-heuristics-for-the-primes-optional/) which discusses *Probabilistic models and heuristics for the primes*. In particular, it gives examples of the use of Cramer's model, while covering much more. I quote:



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> .. we do have a number of extremely convincing and well supported models for the primes (and related objects) that let us predict what the answer to many prime number theory questions (both multiplicative and non-multiplicative) should be, particularly in asymptotic regimes where one can work with aggregate statistics about the primes, rather than with a small number of individual primes.
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> These models are based on taking some statistical distribution related to the primes (e.g. the primality properties of a randomly selected $k$-tuple), and replacing that distribution by a model distribution that is easy to compute with (e.g. a distribution with strong joint independence properties). One can then predict the asymptotic value of various (normalised) statistics about the primes by replacing the relevant statistical distributions of the primes with their simplified models.
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> In this non-rigorous setting, many difficult conjectures on the primes reduce to relatively simple calculations; for instance, all four of the (still unsolved) Landau problems may now be justified in the affirmative by one or more of these models. Indeed, the models are so effective at this task that analytic number theory is in the curious position of being able to confidently predict the answer to a large proportion of the open problems in the subject, whilst not possessing a clear way forward to rigorously confirm these answers!"
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Then comes a warning:



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> In particular, and in contrast to the other notes in this course, the material here is not directly used for proving further theorems, which is why we have marked it as “optional” material. Nevertheless, the heuristics and models here are still used indirectly for such purposes, for instance by
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> giving a clearer indication of what results one expects to be true, thus guiding one to fruitful conjectures
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I will only add two or three concrete observations from the rest of Tao's blogpost.


First, Tao demonstrates that the "naive Cramer model" can recover the Riemann Hypothesis in the form:


For any fixed $\varepsilon>0,$ one has
$$
\sum\_{n\leq x} \Lambda(n)=x+O\_{\varepsilon}(x^{1/2+\varepsilon})
$$
for $x>1.$ Here $\Lambda(n)$ is the von Mangoldt function.


Then, he points out that the naive Cramer model does not predict the constant $e^{\gamma}$ in the third Mertens theorem.


Further down, he shows that naive Cramer yields the conjecture
$$
G(X)=(1+o(1))\log^2 X,
$$
as $X\rightarrow \infty,$ where $G(X)$ is the largest prime gap in $[1,X].$
This conjecture is unproved but to the best of my knowledge the following are the best unconditional bounds on $G(X)$ as $X$ gets large:
$$
 \log X \frac{\log\_2 X \log\_4 X}{\log\_3 X} \ll g(X) \ll \frac{X^{0.525}}{\log X}
$$
where $\log\_2(\cdot)=\log \log((\cdot))$, etc. See another blogpost by Terry Tao  

[here](https://terrytao.wordpress.com/2014/12/16/long-gaps-between-primes/) for the details.


Finally, Tao also illustrates that using the naive Cramer model for a twin prime asymptotic gives the expected result, but that the same model also predicts an infinite number of primes of the form $(p,p+1)$! The reason is that the set of actual primes are not equidistributed modulo 2 but the naive model assumes this.


He then goes on to extensions of the model, but I will stop here.


