Prerequisites for this post: definition of Dirichlet convolution, and big -notation.
Normally I don’t like to blog about something until I’m pretty confident that I have a reasonably good understanding of what’s happening, but I desperately need to sort out my thoughts, so here I go\dots
One day, an alien explorer lands on Earth in a 3rd grade classroom. He hears the teacher talk about these things called primes. So he goes up to the teacher and asks “how many primes are there less than ?”.
Answer: “uh. . .”.
Maybe that’s too hard, so the alien instead asks “about how many primes are there less than ?”.
This is again greeted with silence. Confused, the alien asks a bunch of the teachers, who all respond similarly, but then someone mentions that in the last couple hundred years, someone was able to show with a lot of effort that the answer was pretty close to
The alien, now more satisfied, then says “okay, great! How good is this estimate?”
2. The von Mangoldt function
The prime counting function isn’t very nice, but there is a related function that’s a lot more well-behaved. We define the von Mangoldt function by
It’s worth remarking that in terms of (Dirichlet) convolution, we have
(here is the constant function that gives ). (Do you see why?) Then, we define the second Chebyshev function as
In words, adds up logs of prime powers; in still other words, it is the partial sums of .
It turns out that knowing well gives us information about the number of primes less than , and vice versa. (This is actually not hard to show; try it yourself if you like.) But we like the function because it is more well-behaved. In particular, it turns out the answer to the alien’s question “there are about primes less than ” is equivalent to “”.
So to satisfy the alien, we have to establish and tell him how good this estimate is.
Actually, what we believe to be true is:
Unfortunately, what we actually know is far from this:
You will notice that this error term is greater than , and this is true even of the more modern estimates. In other words, we have a long way to go.
3. Dirichlet Series and Perron’s Formula
Note: I’m ignoring issues of convergence in this section, and will continue to do so for this post.
First, some vocabulary. An arithmetic function is just a function .
The partial sums of an arithmetic function are sums like , or better yet .
Back to the main point. We are scared of the word “prime”, so in estimating we want to avoid doing so by any means possible. In light of this we introduce the Dirichlet series for an arithmetic function , which is defined as
for complex numbers . This is like a generating function, except rather than ‘s we have ‘s.
Why Dirichlet series over generating functions? There are two reasons why this turns out to be a really good move. The first is that in number theory, we often have convolutions, which play well with Dirichlet series:
This is actually immediate if you just multiply it out!
We want to use this to get a handle on the Dirichlet series for . As remarked earlier, we have
The Dirichlet series of has a name; it is the infamous Riemann zeta function, given by
What about ? Answer: it’s just ! This follows by term-wise differentiation of the sum , since the derivative of is .
Thus we have deduced
That was fun. Why do we care, though?
I promised a second reason, and here it is: Surprisingly, complex analysis gives us a way to link the Dirichlet series of a function with its partial sums (in this case, ). It is the so-called \beginPerron’s Formula}, which links partial sums to Dirichlet series:
Applied here this tells us that if is not an integer we have
for any .
This is fantastic, because we’ve managed to get rid of the sigma sign and the word “prime” from the entire problem: all we have to do is study the integral on the right-hand side. Right?
Ha, if it were that easy. That function is a strange beast.
4. The Riemann Zeta Function
Here’s the initial definition:
is the Dirichlet series of the constant function . Unfortunately, this sum only converges when the real part of is greater than . (For , it is the standard harmonic series, which explodes.)
However, we can use something called \beginAbel summation}, which transforms a Dirichlet series into an integral of its partial sums.
It’s the opposite of Perron’s Formula earlier, which we used to transform partial sums into integrals in terms of the Dirichlet series. Unlike , whose partial sums became the very beast we were trying to tame, the partial sums of are very easy to understand:
It’s about as nice as can be!
Applying this to the Riemann zeta function and doing some calculation, we find that
where is the fractional part. It turns out that other than the explosion at , this function will converge for any whose real part is . So this extends the Riemann zeta function to a function on half of the complex plane, minus a point (i.e. is a meromorphic function with a single pole at ).
5. Zeros of the Zeta Function
Right now I’ve only told you how to define for . In the next post I’ll outline how to push this even further to get the rest of the zeta function.
You might already be aware that the behavior of for has a large prize attached to it. For now, I’ll mention that
Proof: Let be the real/imaginary parts (these letters are due to tradition). For , we use the fact that we have an infinite product
Using the fact that , , converges to some finite value, say . By standard facts on infinite products (for example, Appendix A.2 here) that means is .
The situation for is trickier. We use the following trick:
for all . By looking term-by-term at the real parts and using the 3-4-1 inequality we obtain
Now suppose was a zero (); let . Then we get a simple pole at , repeated three times. However, we get a zero at , repeated four times. There is no pole at , so the left-hand side is going to drop to zero, impossible. (The key point is the deep inequality .)