(Notations, non-standard concepts, and definitions used commonly in these investigations are detailed in this post.)

A: The twin prime conjecture

In the post Notes on the Bombieri asymptotic sieve on his blog What’s new, 2006 Fields Medal awardee and Professor of Mathematics at UCLA, Terence Tao, writes and comments that:

“The twin prime conjecture, still unsolved, asserts that there are infinitely many primes $p$ such that $p+2$ is also prime. A more precise form of this conjecture is (a special case) of the Hardy-Littlewood prime tuples conjecture, which asserts that:

$\sum_{n \leq x} \Lambda(n) \Lambda(n+2) = (2\Pi_2+o(1)) x$ … (1)

as $x \rightarrow \infty$, where $\Lambda$ is the von Mangoldt function and $\Pi_2 = 0.6606\dots$ is the twin prime constant:

$\prod_{p>2} (1 - \frac{1}{(p-1)^2})$.

Because $\Lambda$ 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 $\sum_{n \leq x} \Lambda(n) \Lambda(n+2)$ (though this sum will of course have to go to infinity as $x \to \infty$ 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 $\pi(n)$ for finite values of $n$ are apparently derived from real-valued functions which are asymptotic to $\pi(x)$, such as $\frac{x}{log_{e}x}$, $Li(x)$ and Riemann’s function $R(x) = \sum_{n=1}^{\infty}\frac{\mu(n)}{(n)}li(x^{1/n})$.

2. The degree of approximation for finite values of $n$ 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 $\pi(n)$.

3. Moreover, currently, conventional approaches to evaluating prime counting functions for finite $n$ may also subscribe to the belief:

(i) either—explicitly (see here)—that whether or not a prime $p$ divides an integer $n$ is not independent of whether or not a prime $q \neq p$ divides the integer $n$;

(ii) or—implicitly (since the twin-prime problem is yet open)—that a proof to the contrary must imply that if $P(n\ is\ a\ prime)$ is the probability that $n$ is a prime, then $\sum_{_{i = 1}}^{^{\infty}} P(i\ is\ a\ prime) = 1$.

4. If so, then conventional approaches seem to conflate the two probabilities:

(i) The probability $P(a)$ of selecting a number that has the property of being prime from a given set $S$ of numbers;

Example 1: I have a bag containing $100$ 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 $P(b)$ of determining that a given integer $n$ is prime.

Example 2: I give you a $5$-digit combination lock along with a $10$-digit number $n$. The lock only opens if you set the combination to a proper factor of $n$ which is greater than $1$. 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 $S$ is definable, then clearly $P(a)$ too is definable.

However if $S$ is the set $N$ of all integers, and we cannot define a precise ratio of primes to composites in $N$, but only an order of magnitude such as $O(\frac{1}{log_{_{e}}n})$, then equally obviously $P(a)$ cannot be defined in $N$ (see Chapter 2, p.9, Theorem 2.1, here).

6. In case 4(ii) it follows that $P(b) = \frac{1}{\pi(\sqrt{n})}$, since we can show (see Corollary 2.9 on p.14 of this investigation) that whether or not a prime $p$ divides a given integer $n$ is independent of whether or not a prime $q \neq p$ divides $n$.

7. We thus have that $\pi(n) \approx n.\prod_{_{i = 1}}^{^{\pi(\sqrt{n})}}(1-\frac{1}{p_{_{i}}})$. Hence, even though we cannot define the probability $P(n\ is\ a\ prime)$ of selecting a number from the set $N$ of all natural numbers that has the property of being prime, $\prod_{_{i = 1}}^{^{\pi(\sqrt{n})}}(1-\frac{1}{p_{_{i}}})$ can be treated as the de facto probability that a given $n$ is prime.

8. Moreover, by considering the asymptotic density of the set of all integers that are not divisible by the first $\pi(\sqrt {n})$ primes $p_{_{1}}, p_{_{2}}, \ldots, p_{_{\pi(\sqrt {n})}}$ we can show that, for any $n$, the expected number of such integers in any interval of length $(p_{_{\pi(\sqrt{ n})+1}}^{2} - p_{_{\pi(\sqrt n)}}^{2})$ is:

$(p_{_{\pi(\sqrt{ n})+1}}^{2} - p_{_{\pi(\sqrt n)}}^{2})\prod_{i = 1}^{\pi(\sqrt{n})}(1 - \frac{1}{p_{_{i}}})$.

9. We can then show that a non-heuristic approximation for the number of primes less than or equal to $n$ is given for all $n$ by:

$\pi(n) \approx \sum_{j = 1}^{n}\prod_{i = 1}^{\pi(\sqrt{j})}(1 - \frac{1}{p_{_{i}}}) \sim a.\frac{n}{log_{e}n} \rightarrow \infty$

for some constant $a > 2.e^{-\gamma} \approx 1.12292 \ldots$.

10. We can show, similarly, that the expected number of Dirichlet and twin primes in the interval ($p_{_{\pi(\sqrt {n})}}^{2},\ p_{_{\pi(\sqrt{ n})+1}}^{2}$) can be estimated similarly; and conclude that the number of such primes $\leq n$ is, in each case, cumulatively approximated non-heuristically by a function that $\rightarrow \infty$.

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:

$\sum_{_{n \leq x}}\Lambda(n)\Lambda(n+2)$

heuristically by analytic considerations, one could also estimate the number $\pi_{_{2}}(x)$ of twin primes $\leq x$ non-heuristically as:

$\sum_{_{n \leq x}}P(n).P(n+2)$

where $P(n)$ is the de facto probability that a given $n$ 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 $n$ as a $\mathbb{TW}(k)$ integer if, and only if, $r_{p_{_{i}}}(n) \neq 0$ and $r_{p_{_{i}}}(n) \neq 2$ for all $1 \leq i \leq k$, where $0 \leq r_{_{i}}(n) \leq i-1$ is defined for all $i \geq 0$ by:

$n + r_{_{i}}(n) \equiv 0\ (mod\ i)$

3. Note that if $n$ is a $\mathbb{TW}(k)$ integer, then both $n$ and $n+2$ are not divisible by any of the first $k$ primes $\{p_{_{1}}, p_{_{2}}, \ldots, p_{_{k}}\}$.

4. The asymptotic density of $\mathbb{TW}(k)$ integers over the set of natural numbers is then:

$\mathbb{D}(\mathbb{TW}(k)) = \prod_{i=2}^{k}(1 - \frac{2}{p_{_{i}}})$.

5. Further, if $p_{_{k}}^{2} \leq n \leq p_{_{k+1}}^{2}$ is a $\mathbb{TW}(k)$ integer, then $n$ is a prime and either $n+2$ is also a prime, or $n+2 = p_{_{k +1}}^{2}$.

6. If we define $\pi_{_{\mathbb{TW}(k)}}(n)$ as the number of $\mathbb{TW}(k)$ integers $\leq n$, the expected number of $\mathbb{TW}(k)$ integers in any interval $(a, b)$ is given by:

$\pi_{_{\mathbb{TW}(k)}}(b) - \pi_{_{\mathbb{TW}(k)}}(a) \approx (b-a)\prod_{i=2}^{k}(1 - \frac{2}{p_{_{i}}})$

7. Since $\pi_{_{\mathbb{TW}(k)}}(p_{_{k+1}}^{2}) - \pi_{_{\mathbb{TW}(k)}}(p_{_{k}}^{2})$ is at most one less than the number of twin-primes in the interval $(p_{_{k+1}}^{2} - p_{_{k}}^{2})$, it follows that:

$\pi_{_{\mathbb{TW}(k)}}(p_{_{k+1}}^{2}) - \pi_{_{\mathbb{TW}(k)}}(p_{_{k}}^{2}) + 1 \geq \pi_{_{2}}(p_{_{k+1}}^{2}) - \pi_{_{2}}(p_{_{k)}}^{2}) \geq \pi_{_{\mathbb{TW}(k)}}(p_{_{k+1}}^{2}) - \pi_{_{\mathbb{TW}(k)}}(p_{_{k}}^{2})$

8. Now, the expected number of $\mathbb{TW}(k)$ integers in the interval $(p_{_{k+1}}^{2} - p_{_{k}}^{2})$ is given by:

$\pi_{_{\mathbb{TW}(k)}}(p_{_{k+1}}^{2}) - \pi_{_{\mathbb{TW}(k)}}(p_{_{k}}^{2}) \approx (p_{_{k+1}}^{2} - p_{_{k}}^{2})\prod_{i=2}^{k}(1 - \frac{2}{p_{_{i}}})$.

9. We conclude that the number $\pi_{_{2}}(p_{_{k+1}}^{2})$ of twin primes $\leq p_{_{k+1}}^{2}$ is given by the cumulative non-heuristic approximation:

$\sum_{j=1}^{k} (\pi_{_{2}}(p_{_{j+1}}^{2}) - \pi_{_{2}}(p_{_{j}}^{2})) = \pi_{_{2}}(p_{_{k+1}}^{2}) \approx \sum_{j=1}^{k} (p_{_{j+1}}^{2} - p_{_{j}}^{2})\prod_{i=2}^{j}(1 - \frac{2}{p_{_{i}}}) \rightarrow \infty$.

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