Prime Number Theorem

So, I lied. I decided not to give a proof of the prime number theorem. Here’s a brief overview of what it says (without proof). If \pi(x) denotes the number of primes less than or equal to x, then the prime number theorem says

\pi(x)\sim \frac x{\log x}

In other words, if I asked you to estimate the number of primes less than 10^6, a good guess would be 10^6/\log(10^6)\approx 72382.4. In actuality, the number is 78498, so we’d be off by about six-thousand. This may seem like a lot, but compared to the one-million, it’s not that bad. Only 0.6\% error. In fact, here’s a table that might be of interest:

When we say \pi(x)\sim\frac x{\log x}, we essentially mean that as x gets big, the percent error drops to zero (which we see numerically in the table). More technically, we mean that

\displaystyle\lim_{x\to\infty}\frac{\pi(x)\log x}x=1.

So that about wraps up what I want to say about primes. I have no idea what’s coming next, but stay tuned!

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