Not only optimal, but intended.
Caveat: I’m lazy, so grains of salt should be on hand while reading this blog. Due diligence and researching precedents are not my fortes. I depend on google for that, but I do not always avail myself of its services. If you’ve read my previous blogs, then you’re likely already prepared.
Let’s recap. These are a few of my favorite mathematical things:
Ⓐ Primes numbers;
➁︎ Parallelizable spheres in dimensions 1,2,4,8;
Ⅲ︎ Pretty pretty laminated lattices in dimensions 1,2,8,24.
In this episode I want to discuss prime numbers, and how I perceive my thinking about primes differs from – well, this is unclear. Let’s just say I’ve not encountered views similar to those I’m about to extoll, but then, this is not my field. I am merely an enthusiastic dilettante, like a puppy at the beach.
So, I recently happened upon an article suggesting that mathematicians will never stop finding novel ways to prove the prime number theorem. Which is what? Well, if you plot π(n) (the number of primes less than or equal to n) vs n, the plot rises in a manner that looks like it may relate to ln(n). This gave rise to the following approximation:
π(n) ~ n / ln(n).
This is not a very good approximation. I don’t care if the step function π(n) and the smooth function [n / ln(n)] cross each other an infinite number of times (I don’t know that they do), if you look at the table here comparing these two functions, it is clear that Nature is largely uninterested in the correspondence. Still, the ratio of these two functions has been proven to converge to 1, and that is the prime number theorem, which, according to that aforementioned article, is a thing that mathematicians make a hobby of repeatedly proving. Never mind that they diverge in an arithmetic sense; but how about that ratio!
Still, better approximations abound. Maybe [1 / ln(x)] is the density of primes at x, in which case the integral of that from 2 to n, denoted Li(n), should be a good approximation to π(n). In fact, this is a better approximation, but while it is true that π(n) / Li(n) converges to 1, Li(n) does not at all behave like it is Nature’s intended smooth approximation to π(n).
Does Nature have intended smooth approximations to number theory step functions? Is this a thing? Yes, it is. Never mind how I know; just take my word for it.
Evidently Riemann found an exact form for π(n), which is discussed here. Riemann was there first for quite a bit of modern mathematics, and the zeros of his famous zeta function play a part in his exact form. As this involves the as yet unproven Riemann Hypothesis, I’m going to pretend Riemann never existed and carry on discussing π(n) from my own peculiar perspective.
To begin with, there is a problem with the notion of just counting primes as they pop up. What you really want is a measure of prime occurrences. I mean, consider the integers from 1 to 10. There are 4 primes in that collection: 2, 3, 5, 7. But there are more occurrences of primes than just those primes. What I mean by that is this: the prime 2 occurs 3 times, because 2³ = 8 is less than 10. And the prime 3 occurs twice, because 3² = 9 is less than 10. But how do we measure that? In my puppy on a beach investigations I decided upon lcm(n) (least common multiple of the integers from 1 to n). So, for example, lcm(10) = 2³ x 3² x 5 x 7 = 2520.
If you plot lcm(n) it looks a lot like a step function version of en. Did you see? Therefore without a shadow of a doubt Nature’s intended smooth approximation to ln[lcm(n)] is a simple linear function:
ln[lcm(n)] ~ n-1.
Why subtract 1? Well, I did that to enforce equality at n = 1. Nature concurred.
Want another test? Well, if
lcm(n) ~ en-1,
[lcm(n)]1/(n-1) ~ e.
Plot that. Take your time.
Anyway, I will go to my grave (in the not too distant future, unfortunately) fully convinced that these smooth approximations of these number theory step functions are Nature’s intended approximations. And I just don’t understand why in all my beach puppy readings in number theory – back in my beach puppy days – I never encountered this sort of thing. I’m not the only person to have noticed this. If I ever write a book entitled Number Theory for Complete Beach Puppies, this lcm(n) stuff will form its core.
Well, so I used this Nature’s intended approximation to derive a really really good (Nature’s intended) approximation to π(n). And that’s my story. You’ll find that I will be sticking to it like a limpet on a rock.