(Notations, non-standard concepts, and definitions used commonly in these investigations are detailed in this post.)
A: The twin prime conjecture
“The twin prime conjecture, still unsolved, asserts that there are infinitely many primes such that is also prime. A more precise form of this conjecture is (a special case) of the Hardy-Littlewood prime tuples conjecture, which asserts that:
as , where is the von Mangoldt function and is the twin prime constant:
Because is almost entirely supported on the primes, it is not difficult to see that (1) implies the twin prime conjecture.
One can give a heuristic justification of the asymptotic (1) (and hence the twin prime conjecture) via sieve theoretic methods. …
It may also be possible that the twin prime conjecture could be proven by non-analytic means, in a way that does not lead to significantly new estimates on the sum (though this sum will of course have to go to infinity as if the twin prime conjecture holds).”
B: Why heuristic approaches to the twin prime conjecture may not suffice
1. What seems to prevent a non-heuristic determination of the limiting behaviour of prime counting functions is that the usual approximations of for finite values of are apparently derived from real-valued functions which are asymptotic to , such as , and Riemann’s function .
2. The degree of approximation for finite values of is thus determined only heuristically, by conjecturing upon an error term in the asymptotic relation that can be seen to yield a closer approximation than others to the actual values of .
3. Moreover, currently, conventional approaches to evaluating prime counting functions for finite may also subscribe to the belief:
(i) either—explicitly (see here)—that whether or not a prime divides an integer is not independent of whether or not a prime divides the integer ;
(ii) or—implicitly (since the twin-prime problem is yet open)—that a proof to the contrary must imply that if is the probability that is a prime, then .
4. If so, then conventional approaches seem to conflate the two probabilities:
(i) The probability of selecting a number that has the property of being prime from a given set of numbers;
Example 1: I have a bag containing numbers in which there are twice as many composites as primes. What is the probability that the first number you blindly pick from it is a prime. This is the basis for setting odds in games such as roulette.
(ii) The probability of determining that a given integer is prime.
Example 2: I give you a -digit combination lock along with a -digit number . The lock only opens if you set the combination to a proper factor of which is greater than . What is the probability that the first combination you try will open the lock. This is the basis for RSA encryption, which provides the cryptosystem used by many banks for securing their communications.
5. In case 4(i), if the precise proportion of primes to non-primes in is definable, then clearly too is definable.
However if is the set of all integers, and we cannot define a precise ratio of primes to composites in , but only an order of magnitude such as , then equally obviously cannot be defined in (see Chapter 2, p.9, Theorem 2.1, here).
6. In case 4(ii) it follows that , since we can show (see Corollary 2.9 on p.14 of this investigation) that whether or not a prime divides a given integer is independent of whether or not a prime divides .
7. We thus have that , with a binomial standard deviation. Hence, even though we cannot define the probability of selecting a number from the set of all natural numbers that has the property of being prime, can be treated as the de facto probability that a given is prime.
8. Moreover, by considering the asymptotic density of the set of all integers that are not divisible by the first primes we can show that, for any , the expected number of such integers in any interval of length is:
9. We can then show that a non-heuristic approximation—with a binomial standard deviation—for the number of primes less than or equal to is given for all by:
for some constant .
10. We can show, similarly, that the expected number of Dirichlet and twin primes in the interval () can be estimated similarly; and conclude that the number of such primes is, in each case, cumulatively approximated non-heuristically by a function that .
11. The method can, moreover, be generalised to apply to a large class of prime counting functions (see Section 5.A on p.22 of this investigation).
C: A non-heuristic proof that there are infinitely many twin primes
1. In particular, instead of estimating:
heuristically by analytic considerations, one could also estimate the number of twin primes non-heuristically as:
where is the de facto probability that a given is prime; and where we need to allow for the possibility that the two probabilities may not be independent.
2. One way of approaching this would be to define an integer as a integer if, and only if, and for all , where is defined for all by:
3. Note that if is a integer, then both and are not divisible by any of the first primes .
4. The asymptotic density of integers over the set of natural numbers is then:
5. Further, if is a integer, then is a prime and either is also a prime, or .
6. If we define as the number of integers , the expected number of integers in any interval is given—with a binomial standard deviation—by:
7. Since is at most one less than the number of twin-primes in the interval , it follows that:
8. Now, the expected number of integers in the interval is given by:
9. We conclude that the number of twin primes is given by the cumulative non-heuristic approximation: