Can the Probability of a Number Being a Product of Primes be Calculated?

[PeterJ1]

I am not a mathematician but enjoy studying the primes as a mechanical system. Regrettably I don't have the tools (and perhaps the intellect) for some jobs.

I've spent some time trying to calculate the probability of a number being a product of each successive prime as the number line is progressively sieved, and have decided it cannot be done. But maybe this is just ignorance.

It is simple for 2,3, and 5.
Sieve out the products of 2 and 1/3 of the remainder will be products of 3.
Sieve out the products of 3 and 1/5 of the remainder will be products of 5.
Sieve out the products of 5 and about 1/7.5 of the remainder...

From here on the proportion become increasingly difficult to calculate.

Is this calculation possible? Or must it always be a statistical approximation?

 

[Petek]

I think what you're looking for is what's known as the Inclusion-exclusion principle.

 

[topsquark]

I'm tempted to create a new user account with the name PeteL...

-Dan

 

[PeterJ1]

Ha.

Thanks Petek. The inc-exc approach does seem relevant but I wouldn't know how it can be used in this situation.

I suspect that the calculation cannot be done directly but that a result can be approximated using the PNT. But I'm not sure about this. Now I look at it I wonder if I'm just trying to re-create the PNT. I thought coming at it from this angle would be managable but as usual it's the endless error corrections that defeat me.

Suppose for some defined region we sieved out all the primes up to 13. Would it then be possible to calculate the probability that a number in this region is a product of 17 without using the PNT?

 

[HallsofIvy]

I am not a mathematician but enjoy studying the primes as a mechanical system. Regrettably I don't have the tools (and perhaps the intellect) for some jobs.

I've spent some time trying to calculate the probability of a number being a product of each successive prime as the number line is progressively sieved, and have decided it cannot be done. But maybe this is just ignorance.

It is simple for 2,3, and 5.
Sieve out the products of 2 and 1/3 of the remainder will be products of 3.
Sieve out the products of 3 and 1/5 of the remainder will be products of 5.
Sieve out the products of 5 and about 1/7.5 of the remainder...

From here on the proportion become increasingly difficult to calculate.

Is this calculation possible? Or must it always be a statistical approximation?

What probability distribution are you assuming? You cannot have a uniform probability distribution with an infinite set.

 

[PeterJ1]

What probability distribution are you assuming? You cannot have a uniform probability distribution with an infinite set.

Hi HallsofIvy -Thanks for replying.

I'm not assuming anything at all (I hope). I'm wondering whether it is possible to calculate the changing probability of N being a product of a prime as the products of each prime in turn are sieved out.

This is easy for the primes 2.3 and 5. This leaves 8 in 30 numbers unsieved.

There would now be 1/7 products of 7 among the numbers that remain, but a correction would be requires for joint products of 5 and 7 (which have already been sieved).

Then for 11, three or four correction terms would be required. And so on.

So, as the products of each prime is sieved in turn the chances of N being a product of the next prime will become ever further from 1/p.

The calculation seems to end up being the same as it would be distribution of primes around N but comes at it from a slightly different direction.

My most naive question would be: Are these correction terms playing the same role as Riemann's correction terms?

If this is an idiotic question I wouldn't be surprised.

 

[PeterJ1]

Bump

This seems a naive question but a reasonable one and I really would like to hear a response from a mathematician. You could think of it as charity work with the mathematically impaired. :)