I am wondering if my approach below, based on probability theory and thus heuristic (it is by no means a proof) is new and worth pursuing. Applications are numerous.

Let $S_f$ be a random set of positive integers defined as follows: the positive integer $n$ belongs to $S_f$ with probability $f(n)$. In other words, for each integer $n$, generate a uniform deviate $U_n$ on $[0, 1]$. If and only if $U_n < f(n)$ then add $n$ to $S_f$. The deviates $U_n$ are independent. Now consider $f(n) = \frac{1}{\log n}$ if $n > 2$, and $f(1)= f(2) = 0$. With this particular choice of $f$, the set $S_f$ has the same density as the set of prime numbers, and behaves in a very similar way. Let us define the following functions (actually, they are random variables):

  • $M(n)$ is the number of elements in $S_f$ that are smaller than $n$.

  • $T(n)$ is the number of twin pairs $(k, k+1)$ with both $k, k+1 \in S_f$ and $k \leq n$.

The twins in $S_f$ play the same role as twin primes. It is is easy to prove that $M(n)$ (its expectation) is asymptotically equal to the logarithm integral $\mbox{Li}(n)$. The same is true for the prime number counting function. Furthermore, the prime numbers themselves is a particular case of $S_f$, a particular realization of the random set $S_f$ for the same function $f$. (you need to ignore even integers in the construction process, but other than that, it's straightforward.)

Then we also have: $$\mbox{E}[T(n)] \sim C\sum_{k=1}^{n-1} f(k)f(k+1)\sim C'\int_2^n \frac{dt}{(\log t)^2} .$$

Here E denotes the expectation. The same result is conjectured to be true for twin primes. It is easy to show that there are infinitely many twin numbers in $S_f$, and assess whether the series consisting of the reciprocals of these twins, converge. The same arguments could be applied to twin primes, but of course, it would no constitute a proof, just some heuristic argumentation. Are there any useful references related to the approach discussed here?

Update: I plan on using the same methodology for the Goldbach conjecture, and in particular to obtain an asymptotic value for the expected number of ways an even number $n$ can be represented as a sum of two primes. With my notation, it should be asymptotically equivalent to $$\sum_{k=1}^{\lfloor n/2 \rfloor} f(k)f(n-k) \sim \frac{n}{2(\log n)^2}.$$ See also here and here.


1 Answer 1


You're about 90 years too late. The probabilistic approach to prime numbers was developed by Cramér in the 1930's. You might look at this paper by the other Granville.

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    $\begingroup$ I am writing a book on probabilistic number theory, I don't want to mention something is new if it is not, thus the reason for my question. The Wikipedia entry (see en.wikipedia.org/wiki/Twin_prime) on Twin Primes mentions that " The conjecture can be justified (but not proven) by assuming that 1 / ln t describes the density function of the prime distribution, an assumption suggested by the prime number theorem and would imply the twin prime conjecture, but remains unresolved." but with no references and no explanation. $\endgroup$ Sep 29, 2019 at 2:01
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    $\begingroup$ Here is a PDF version of that document: dartmouth.edu/~chance/chance_news/for_chance_news/Riemann/… $\endgroup$ Sep 29, 2019 at 2:10

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