Harristotelian Logic

I have been talking for a bit about how simply noting that given a natural x and a simple result about primes, you can talk about prime probability. In this post I'll show how powerful an idea that is by relating it to the prime number theorem and by using Mertens's Theorem.

First off the idea is that given primes p_1 and p_2, there is no preference for

p_1 mod p_2

for any particular non-zero residue of p_2, so that when having some natural x, the probability that x does NOT have p_2 as a factor if p_2 is less than or equal to sqrt(x) is roughly 1 - 1/p, which is (p-1)/p.

Then I can express the probability that a natural x is prime by

probPrime(x) = ((p_j - 1)/p)*···(1/2)

where there are j primes less than or equal to sqrt(x) and p_j is the jth prime.

Now I'll compare results from this idea with what follows from the prime number theorem, and I'll use small examples because they're easier, while it's trivial for others to extend out indefinitely to satisfy themselves and others.

I will use x = 10, x=100 and x=1000.

At x = 10, I have probability of primeness as (2/3)*(1/2) = 1/3, and 10/3 is approximately 3.3.

There are 4 primes in the interval, so 4 - 3 = 1.

Here using the Wikipedia prime number theorem article as a reference.

Li(10) = pi(10) = 2.2

At x=100, I have for my result:

(6/7)*(4/5)*(2/3)*(1/2) = 8/35, and 100(8/35) is approximately 22.9.

There are 25 primes in the interval, so 25 - 22.9 = 2.1.

Li(100) - pi(100) = 5.1.

Finally, at x=1000, I have for my result:

(30/31)*(28/29)*(22/23)*(18/19)*(16/17)*(12/13)*(10/11)*(6/7)*(4/5)*(2/3)*(1/2)

which is approximately 0.153 to three significant figures, so times 1000, I have 153.

There are 168 primes so 168 - 153 = 15.

Li(1000) - pi(1000) = 10.

So it's further off, but not by much. That's as much as I'm going to do by hand. I wonder if anyone else might step in and program this thing and step it out?

Just for fun, how does my idea compare against x/(ln x)?

At x=10, 10/(ln 10) is about 4.3, so it is only about 8% off.

While my idea with 3.3 is about 18% off.

At x=100, 100/(ln 100) is about 21.7, so it is about 13% off.

While my idea with 22.9 is about 8% off.

At x=1000, 1000/(ln 1000) is about 145, so it is about 14% off.

While my idea with 153 is about 9% off.

Mertens's theorem comes into this because as other posters pointed out

((p_j - 1)/p)*···(1/2) approx equals (2*e^{-gamma))/(ln x)

when p_j is again the largest prime up to or equal to sqrt(x).

If my idea holds, and logically it should, then it follows that

(2*e^{-gamma))/(ln x)

approximates the correct value for the probability of primeness as x goes out to infinity.

And that is, oddly enough, possibly new.

If that connection has not been made until now, then no doubt about it people, you are witnessing history, and I'll give the benefit of the doubt, and wait for either someone to cite showing the connection has been made before, or for mathematicians to show appreciation for a beautiful result.

For amateur mathematicians this could be exciting showing what was available still in the way of simple, yet powerful ideas at the very heart of important areas, like core behavior of prime numbers.