A Formula for Primes

I wanted to write this post yesterday, but I couldn’t work out some of the steps that I wanted to be able to work out, so I had to wait a day and try again. But I think I’ve got it now, let’s see…

We’ve been working with the function

\displaystyle J(x)=\sum_{p^n\leq x}\frac{1}{n}.

Last time, we arrived (skipping over some rather large gaps) at an analytic formula for J(x), namely:

\begin{array}{l}\displaystyle J(x)=Li(x)-\sum_{\text{Im }\rho>0}\left(Li(x^{\rho})+Li(x^{1-\rho})\right)\\ \displaystyle\qquad\qquad +\int_x^{\infty}\frac{dt}{t(t^2-1)\ln t}-\ln 2,\end{array}

where (mod issues)

\displaystyle Li(x)=\int_0^x \frac{dt}{\ln t}.

The goal today is to use this formula for J(x) to find a formula for primes. In particular, we will find a formula for \pi(x), which is the number of primes no larger than x. These two functions are related, as follows: recall that J(x) is a step function which jumps by 1/n at every prime power p^n. Pick an x_0. How many jumps of size 1 does J(x) make before getting to x_0? Well, it jumps by 1 at the primes, so it makes \pi(x_0) jumps of size 1. How many jumps of size 1/2? These are jumps at prime squares, p^2\leq x_0. This inequality is the same as p\leq x_0^{1/2}, so we make \pi(x_0^{1/2}) jumps of size 1/2. Similarly, we’ll make \pi(x_0^{1/n}) jumps of size 1/n to calculate J(x_0). Eventually (for some large enough N) x_0^{1/N} will be less than 2, and so \pi(x_0^{1/N})=0, and we can stop looking for jumps.

Translating the above into symbols, we have just decided that

J(x)=\pi(x)+\frac{1}{2}\pi(x^{1/2})+\frac{1}{3}\pi(x^{1/3})+\cdots+\frac{1}{n}\pi(x^{1/n})+\cdots.

I’ve written it as an infinite sum, but eventually (like I said) those \pi values will be 0, so we’ve really got a finite sum.

Now wait, our goal was a formula for \pi(x), in terms of J(x), and I’ve got a formula the other way, J(x) in terms of \pi(x).

This is the part I got stuck on yesterday. Edwards’ book says (or perhaps doesn’t, causing my problems) that basically you do Möbius inversion to the sum above, and obtain a formula for \pi(x) in terms of J(x). He says slightly more than that, but I wasn’t able to piece it together as easily as he seemed to indicate it should work. But anyway, what’s this Möbius inversion thing? I’ll take a pretty low-level account, leaving lots of room for things to be jazzed up (which I may return to and do sometime soonish).

An integer is said to be “squarefree” if it factors into the product of distinct primes or, equivalently, it is not divisible by a square. Define the following function, called the Möbius function, on positive integers (p_i is supposed to indicate a prime):

\displaystyle \mu(n) = \begin{cases}1& n=1 \\ 0 & n\text{ not squarefree} \\ (-1)^k & n=p_1\cdots p_k\text{ squarefree}\end{cases}

So, for example, \mu is -1 at any prime, \mu(6)=\mu(10)=1, and \mu(8)=\mu(12)=0.

Möbius inversion is then the following:

If f(n)=\sum_{d|n}g(d) then g(d)=\sum_{d|n}\mu(d)f(n/d), and conversely.

Well, I tried to figure out how my expression for J(x), in terms of \pi(x^{1/n})s was such an f of g. I don’t see it, as stated. Perhaps somebody can point me in the right direction in the comments.

The observation you are supposed to make comes when you consider something like \frac{1}{2}J(x^{1/2}). Using our formula above, we can write

\frac{1}{2}J(x^{1/2})=\frac{1}{2}\pi(x^{1/2})+\frac{1}{4}\pi(x^{1/4})+\frac{1}{6}\pi(x^{1/6})+\cdots

or, more generally, we have

\displaystyle \frac{1}{n}J(x^{1/n})=\sum_k \frac{1}{nk}\pi(x^{1/nk}).

If we subtract this equation (with n=2) from the equation for J(x) in terms of \frac{1}{n}\pi(x^{1/n})s, we can eliminate one term from the right-hand side. The idea is to iterate this process. I think a (reasonably, by my blog’s standards) clean way to write what is happening is as follows (it helps when you know the answer you are supposed to get :)):

Consider the sum

\displaystyle \sum_n \frac{\mu(n)}{n}J(x^{1/n}),

which by our most recent formula is the same as

\displaystyle \sum_{n,k}\frac{\mu(n)}{nk}\pi(x^{1/nk}).

Now, fix a positive integer m. The coefficient of \pi(x^{1/m}) in this last double sum is

\displaystyle \sum_{nk=m}\frac{\mu(n)}{m}=\frac{1}{m}\sum_{n|m}\mu(n).

Almost there… what is \sum_{n|m}\mu(n)? Turns out, it is 1 if m=1 (which is clear), and 0 otherwise. Indeed, suppose m has prime factorization p_1^{e_1}\cdots p_k^{e_k}. Then all of the divisors, n in the sum we’re after, are of the form p_1^{f_1}\cdots p_k^{f_k} where each 0\leq f_i\leq e_i. The only divisors, n, where \mu(n)\neq 0 are those where the f_i are all either 0 or 1. So the worthwhile n could be 1, or some p_i, or a p_ip_j (i\neq j), or p_ip_jp_{\ell} (i\neq j, i\neq \ell, j\neq \ell), etc. So

\begin{array}{l}\displaystyle \sum_{n|m}\mu(n)=1+\sum_i \mu(p_i)+\sum_{i,j} \mu(p_ip_j)+\cdots+\mu(p_1\cdots p_k) \\ \qquad \qquad \displaystyle =1-k+\binom{k}{2}-\binom{k}{3}+\cdots+(-1)^k=(1-1)^k=0.\end{array}

(By the way, this is (almost verbatim) the proof in Hardy and Wright’s “An Introduction to the Theory of Numbers”, and I’ll count that as one of my references for anything I say about \mu).

So, let’s gather back up here. We’ve now shown that in the sum

\displaystyle \sum_n \frac{\mu(n)}{n}J(x^{1/n})=\sum_{n,k}\frac{\mu(n)}{nk}\pi(x^{1/nk})

the coefficient of \pi(x^{1/m}) is 0 unless m=1, when the coefficient is 1. Therefore, we have obtained out formula for \pi(x):

\begin{array}{l} \pi(x)=\displaystyle \sum_n \frac{\mu(n)}{n}J(x^{1/n})\\ \qquad \displaystyle=J(x)-\frac{1}{2}J(x^{1/2})-\frac{1}{3}J(x^{1/3})-\frac{1}{5}J(x^{1/5})+\frac{1}{6}J(x^{1/6})-\cdots\end{array}

I guess the point of Riemann’s paper was that the terrible integral formula he obtained for J(x) could then be used in this expression for \pi(x), to give an exact analytic expression (that is, one only an analyst would find pretty) for \pi(x). Pretty wild, in my mind.

It seems that if you get some feeling for the terms in J(x), you can decide that the “dominant” term is the Li(x) term. And then that still ends up as the dominant term in the expression above for \pi(x). And so you get to say

\pi(x)\sim Li(x).

So you’ve got that going for you, which is nice.

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One Response to “A Formula for Primes”

  1. Ray Says:

    interesting

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