Abstract We define the residues for all and all such that if, and only if, is a divisor of . We then show that the joint probability of two unequal primes dividing any integer is the product . We conclude that the prime divisors of any integer are independent; and that the probability of being a prime is . The number of primes less than or equal to is thus given by . We further show that , and conclude that does not oscillate.
We begin by defining the residues for all and all as below:
Definition 1: where .
Since each residue cycles over the values , these values are all incongruent and form a complete system of residues  .
It immediately follows that:
Lemma 1: if, and only if, is a divisor of .
By the standard definition of the probability of an event , we conclude that:
Lemma 2: For any and any given integer , the probability that is , and the probability that is .
We note the standard definition:
Definition 2: Two events and are mutually independent for if, and only if, .
The prime divisors of any integer are mutually independent
We then have that:
Lemma 3: If and are two primes where then, for any , we have:
where and .
Proof: The numbers , where and , are all incongruent and form a complete system of residues  . Hence:
By Lemma 2:
The lemma follows.
If and in Lemma 3, so that both and are prime divisors of , we conclude by Definition 2 that:
Corollary 1: .
Corollary 2: .
Theorem 1: The prime divisors of any integer are mutually independent. 
The probability that is a prime
Since is a prime if, and only if, it is not divisible by any prime , it follows immediately from Lemma 2 and Lemma 3 that:
Lemma 4: For any , the probability of an integer being a prime is the probability that for any if .
Lemma 5: .
The Prime Number Theorem
The number of primes less than or equal to is thus given by:
Lemma 6: .
This now yields the Prime Number Theorem:
Theorem 2: .
Proof: From Lemma 6 and Mertens’ Theorem that
it follows that:
The behaviour of as is then seen by differentiating the right hand side, where we note that :
Hence does not oscillate as .
I am indebted to my erstwhile classmate, Professor Chetan Mehta, for his unqualified encouragement and support for my scholarly pursuits over the past fifty years; most pertinently for his patiently critical insight into the required rigour without which the argument of this 1964 investigation would have remained in the informal universe of seemingly self-evident truths.
HW60 G. H. Hardy and E. M. Wright. 1960. An Introduction to the Theory of Numbers 4th edition. Clarendon Press, Oxford.
Ti51 E. C. Titchmarsh. 1951. The Theory of the Riemann Zeta-Function. Clarendon Press, Oxford.
Return to 1: HW60, p.49.
Return to 2: HW60, p.52, Theorem 59.
Return to 3: In the previous post we have shown how it immediately follows from Theorem 1 that integer factorising is necessarily of order ; from which we conclude that integer factorising cannot be in the class of polynomial-time algorithms.
Return to 4: By HW60, p.351, Theorem 429, Mertens’ Theorem.
Return to 5: By the argument in Ti51, p.59, eqn.(3.15.2).
Return to 6: HW60, p.9, Theorem 7.