# von Mangoldt and Zeta

Prerequisites for this post: definition of Dirichlet convolution, and big ${O}$-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

## 1. Primes

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 ${x}$?”.

Maybe that’s too hard, so the alien instead asks “about how many primes are there less than ${x}$?”.

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

$\displaystyle \approx \frac{x}{\log x}.$

The alien, now more satisfied, then says “okay, great! How good is this estimate?”

More silence.

## 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 ${\Lambda}$ by

$\displaystyle \Lambda(x) = \begin{cases} \log p & x = p^k \text{ for some prime } p \\ 0 & \text{otherwise}. \end{cases}$

It’s worth remarking that in terms of (Dirichlet) convolution, we have

$\displaystyle \mathbf 1 \ast \Lambda = \log$

(here ${\mathbf 1}$ is the constant function that gives ${\mathbf 1(n) = 1}$). (Do you see why?) Then, we define the second Chebyshev function as

$\displaystyle \psi(x) = \sum_{n \le x} \Lambda(n).$

In words, ${\psi(x)}$ adds up logs of prime powers; in still other words, it is the partial sums of ${\Lambda}$.

It turns out that knowing ${\psi(x)}$ well gives us information about the number of primes less than ${x}$, and vice versa. (This is actually not hard to show; try it yourself if you like.) But we like the function ${\psi}$ because it is more well-behaved. In particular, it turns out the answer to the alien’s question “there are about ${\frac{x}{\log x}}$ primes less than ${x}$” is equivalent to “${\psi(x) \approx x}$”.

So to satisfy the alien, we have to establish ${\psi(x) \approx x}$ and tell him how good this estimate is.

Actually, what we believe to be true is:

Theorem 1 (Riemann Hypothesis)

We conjecture that

$\displaystyle \psi(x) = x + O\left( x^{\frac{1}{2}+\varepsilon} \right)$

for any ${\varepsilon > 0}$.

Unfortunately, what we actually know is far from this:

Theorem 2 (Prime Number Theorem, Classical Form)

We have proved that

$\displaystyle \psi(x) = x + O\left( x e^{-c\sqrt{\log x}} \right)$

for some constant ${c}$ (actually we have done slightly better, but not much).

You will notice that this error term is greater than ${O(x^{0.999})}$, 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 ${\mathbb N \rightarrow \mathbb C}$.

Example 3

Functions ${\Lambda}$, ${\mathbf 1}$, or ${\log}$ are arithmetic functions.

The partial sums of an arithmetic function are sums like ${f(1) + f(2) + \dots + f(n)}$, or better yet ${\sum_{n \le x} f(n)}$.

Example 4

The Chebyshev function ${\psi}$ gives the partial sums of ${\Lambda}$, by definition.

Example 5

The floor function ${\left\lfloor x \right\rfloor}$ gives the partial sums of ${\mathbf 1}$:

$\displaystyle \left\lfloor x \right\rfloor = \sum_{n \le x} 1 = \sum_{n \le x} \mathbf 1(n).$

Back to the main point. We are scared of the word “prime”, so in estimating ${\psi(x)}$ we want to avoid doing so by any means possible. In light of this we introduce the Dirichlet series for an arithmetic function ${f}$, which is defined as

$\displaystyle F(s) = \sum_{n \ge 1} f(n) n^{-s}$

for complex numbers ${s}$. This is like a generating function, except rather than ${x^n}$‘s we have ${n^{-s}}$‘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:

Theorem 6 (Convolution of Dirichlet Series)

Let ${f, g, h : \mathbb N \rightarrow \mathbb C}$ be arithmetic functions and let ${F}$, ${G}$, ${H}$ be the corresponding Dirichlet series. Then

$\displaystyle f = g \ast h \implies F = G \cdot H.$

(Here ${\ast}$ is the Dirichlet convolution.)

This is actually immediate if you just multiply it out!

We want to use this to get a handle on the Dirichlet series for ${\Lambda}$. As remarked earlier, we have

$\displaystyle \log = \mathbf 1 \ast \Lambda.$

The Dirichlet series of ${\mathbf 1}$ has a name; it is the infamous Riemann zeta function, given by

$\displaystyle \zeta(s) = \sum_{n \ge 1} n^{-s}.$

What about ${\log}$? Answer: it’s just ${-\zeta'(s)}$! This follows by term-wise differentiation of the sum ${\zeta}$, since the derivative of ${n^{-s}}$ is ${-\log s \cdot n^{-s}}$.

Thus we have deduced

Theorem 7 (Dirichlet Series of von Mangoldt)

We have

$\displaystyle -\frac{\zeta'(s)}{\zeta(s)} = \sum_{n \ge 1} \Lambda(n) \cdot n^{-s}.$

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, ${\psi}$). It is the so-called \beginPerron’s Formula}, which links partial sums to Dirichlet series:

Theorem 8 (Perron’s Formula)

Let ${f : \mathbb N \rightarrow \mathbb C}$ be a function, ${F}$ its Dirichlet series. Then for any ${x}$ not an integer,

$\displaystyle \sum_{n \le x} f(n) = \frac{1}{2\pi i} \int_{c-i\infty}^{c+i\infty} F(s) \cdot \frac{x^s}{s} \; ds$

for any large enough ${c}$ (large enough to avoid convergence issues).

Applied here this tells us that if ${x}$ is not an integer we have

$\displaystyle \psi(x) \overset{\text{def}}{=} \sum_{n \le x} \Lambda(n) = \frac{1}{2\pi i} \int_{c-i\infty}^{c+i\infty} -\frac{\zeta'(s)}{\zeta(s)} \cdot \frac{x^s}{s} \; ds.$

for any ${c > 1}$.

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 ${\zeta}$ function is a strange beast.

## 4. The Riemann Zeta Function

Here’s the initial definition:

$\displaystyle \zeta(s) = \sum_{n \ge 1} n^{-s}$

is the Dirichlet series of the constant function ${\mathbf 1}$. Unfortunately, this sum only converges when the real part of ${s}$ is greater than ${1}$. (For ${s=1}$, 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.

Theorem 9 (Abel Summation for Dirichlet Series)

If ${f}$ is an arithmetic function and ${F}$ is its Dirichlet series then

$\displaystyle F(s) = s \int_1^{\infty} \frac{\sum_{n \le x} f(n)}{x^{s+1}} \; dx.$

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 ${\Lambda}$, whose partial sums became the very beast ${\psi}$ we were trying to tame, the partial sums of ${\mathbf 1}$ are very easy to understand:

$\displaystyle \sum_{n \le x} \mathbf 1(n) = \sum_{n \le x} 1 = \left\lfloor x \right\rfloor.$

It’s about as nice as can be!

Applying this to the Riemann zeta function and doing some calculation, we find that

$\displaystyle \zeta(s) = \frac{s}{s-1} - s \int_1^\infty \frac{ \left\{ x \right\} }{x^{s+1}} \; ds$

where ${\left\{ x \right\}}$ is the fractional part. It turns out that other than the explosion at ${s=1}$, this function will converge for any ${s}$ whose real part is ${> 0}$. 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 ${s=1}$).

## 5. Zeros of the Zeta Function

Right now I’ve only told you how to define ${\zeta(s)}$ for ${\mathrm{Re}\; s > 0}$. 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 ${\zeta}$ for ${0 < \mathrm{Re}\; s < 1}$ has a large prize attached to it. For now, I’ll mention that

Theorem 10

If ${\mathrm{Re}\; s \ge 1}$, then ${\zeta(s) \neq 0}$.

Proof: Let ${s = \sigma + it}$ be the real/imaginary parts (these letters are due to tradition). For ${\sigma > 1}$, we use the fact that we have an infinite product

$\displaystyle \zeta(s) = \prod_{p \text{ prime}} \left( 1+p^s+(p^2)^s + \dots \right) = \prod_{p \text{ prime}} \frac{1}{1-p^{-s}}.$

Using the fact that ${\sum_p \left\lvert p^{-s} \right\rvert < \sum_p p^{-\sigma} < \sum_{n \ge 1} n^{-\sigma} < \infty}$, ${\prod_p (1-p^{-s})}$, converges to some finite value, say ${L}$. By standard facts on infinite products (for example, Appendix A.2 here) that means ${\zeta(s)}$ is ${1/L \neq 0}$.

The situation for ${\sigma = 1}$ is trickier. We use the following trick:

$\displaystyle 3 + 4\cos\theta + \cos2\theta = 2(\cos\theta+1)^2 \ge 0 \quad \forall \theta.$

Now,

$\displaystyle \log\zeta(s) = \sum_p -\log(1-p^{-s}) = \sum_p \sum_{n \ge 1} \frac{p^{-sn}}{n}$

for all ${\sigma > 1}$. By looking term-by-term at the real parts and using the 3-4-1 inequality we obtain

$\displaystyle 3\,\mathrm{Re}\, \log\zeta(\sigma) + 4\,\mathrm{Re}\, \log\zeta(\sigma+it) + \mathrm{Re}\, \log\zeta(\sigma+2it) \ge 0 \qquad \sigma > 1.$

Thus

$\displaystyle \left\lvert \zeta(\sigma)^3\zeta(\sigma+it)^4\zeta(\sigma+2it) \right\rvert \ge 1.$

Now suppose ${1+it}$ was a zero (${t \neq 0}$); let ${\sigma \rightarrow 1^+}$. Then we get a simple pole at ${\zeta(1)}$, repeated three times. However, we get a zero at ${\zeta(1+it)}$, repeated four times. There is no pole at ${\zeta(1+2it)}$, so the left-hand side is going to drop to zero, impossible. (The key point is the deep inequality ${4 > 3}$.) $\Box$

Next up: prime number theorem. References: Kedlaya’s 18.785 notes and Hildebrand’s ANT notes.