I was reading a paper which started with this line:

The Dedekind eta function \(\eta(\tau)\) is defined for \(\tau \in H\) by the infinite product: \[ \eta(\tau) = q^{1/24} \prod_{n=1}^{\infty} (1 – q^n), \quad \text{where } q = e^{2i\pi\tau} \]

I railed against this. Why ?

• Because I have no context. Why would I care about this definition? How do I know it’s important or useful?
• What is 24 doing there?
• It says \(\tau\) belongs to \(H\). But I do not know what \(H\) is?
• Is \(\tau\) a complex value or a real number or what?
• What is the domain and the range of this function \(\eta\)? Is it \(H \rightarrow \mathbb{C}\)?
• And why are we using a product, instead of a more familiar sum?
• And what does \(q=e^{2 \pi i \tau}\) really do there?

Too many questions, too many unknowns. My brain couldn’t hold it. Hence the frustration.

So what do I do in this situation?

One option is to move on and hope that I will understand it later.The knee jerk refles is to do exactly that. But I will choose another approach. I will study this definition and see if I can relate it to my previous knowledge. I will be like a detective and see what clues I can get out of this definition.

But I’m not enrolled in a course now, so I will choose another approach. I will study this definition and see if I can relate it to my previous knowledge. I will be like a detective and see what clues I can get out of this definition.

Let me focus now on the product. What does it mean? I’ll expand the product directly by hand

\((1-q)(1-q^2) = 1 – q – q^2 + q^3\)

\((1-q)(1-q^2)(1-q^3) = 1 – q – q^2 + q^4 + q^5 – q^6\)

Keep going on and I get: \[ P(q)=\prod(1-q^n) = 1 – q – q^2 + q^5 + q^7 – q^{12} – q^{15} + q^{22} + q^{26} – … \]

On the lhs I have a product and on the rhs I have a sum, which is good. I recognize the exponents as the Euler’s pentagonal numbers.

Evaluate for the non-negative integers \[ \frac{k(3k-1)}{2} \rightarrow 0, 1, 5, 12, 22, … \] Evaluate for the negative integers \[ \frac{k(3k-1)}{2} \rightarrow 2, 7, 15, 26, 40, … \]

When does the sum converge?

The non-zero terms are of the form \(q^n=e^{2 \pi i n \tau}\). Set \(\tau=x+iy\). Then \[ q^n=e^{2 \pi i n \tau}=e^{2 \pi i n x} e^{-2 \pi n y}, \qquad |q^n| = e^{-2\pi n y} \] These terms go to zero (pretty fast) when \(y>0\). For \(y≤0\) the terms do not go to to zero. So \(H\) must mean the upper half plane \(H\) = Im(\(\tau\)) > 0 (real axis not included). And on that domain \(\eta\) takes complex values, so it is a map \(\eta: H \rightarrow\mathbb{C}\).

Since this is a complex valued function, the standard is to investigate if it is holomorphic.

Now \(q = e^{2 \pi i \tau}\) is a holomorphic function of \(\tau\). Each finite partial product of \(\eta\) is a polynomial in q, hence holomorphic in \(\tau\) . And because the terms shrink exponentially fast for any fixed \(y>0\), the partial product converge uniformly on compact regions of \(H\) — which is exactly the hypothesis Weierstrass’s theorem needs to guarantee the limit function \(\eta\) is holomorphic too.

Looking at \(q^n=e^{2 \pi i n \tau}\), I see that it is periodic because \(e^{2 \pi i n (\tau + 1)} = e^{2 \pi i n \tau}\). what about \(\eta( \tau)\) ?

The function \(\eta\) is not just the product — there is a factor \(q^{1/24}\) in front. So \[ \eta(\tau+1)=e^{2 \pi i/24}\,\eta(\tau) \] \(\eta\) is not periodic; it picks up a 24th root of unity.

What does the map \(\tau \mapsto q = e^{2\pi i \tau}\) sends the upper half plane onto?

\(q\) is a variable created by the formula \(\tau \mapsto q = e^{2\pi i \tau} = e^{2\pi i x} e^{-2\pi y}\) , where \(\tau=x+iy\) is a complex number. I can think of it as a change of coordinates — I’m converting from \(\tau\) -space to \(q\)-space. Take any point \(\tau\) in the upper half plane \(H\) and fix \(y\). So any point along a horizontal line will be transformed onto a point on a circle with radius \(e^{-2 \pi y}\). Furthermore \(q(\tau+1)= q(\tau)\) means that as I travel along that line the x values that that differ by an integer will produce identical q values. So the map wraps the horizontal line from the \(\tau\)-plane around the circle infinitely many times in the \(q\)-plane.

Now what happens if we let \(y \rightarrow \infty\). Then \(|q|=e^{-2 \pi y} \rightarrow 0\). So we are moving towards the origin.

Now what happens if we let \(y \rightarrow 0\). Then \(|q|=e^{-2 \pi y} \rightarrow 1\). So we are moving towards the unit circle. So overall H is mapped onto a punctured open unit disc.

What happens if we hold \(x\) fixed and move along a vertical line. Then \(e^{2 \pi i x}\) determines a fix angle, and y varies so \(e^{-2 \pi y}\) will assume values between 1 and 0. So we get radial rays.

So the map \(\tau \mapsto q = e^{2\pi i \tau}\) sends the upper half plane onto the punctured unit disk \(0 < |q| < 1\). So instead of looking \(\eta\) as a function of \(\tau\) on the upper half plane, we can look at it as a function on \(q\) defined on the punctured unit disc. That answers the question of what \(q\) was doing there.

So I have answered some questions but not these ones:

• Why would I care about this definition? How do I know it’s important or useful?
• What is 24 doing there?

Let’s see if I can figure out the number 24 properly. In the expansion above I had the pentagonal numbers generated by \(\frac{k(3k-1)}{2}\) where \(k\) took on both positive and negative integers.

So I see that the product can be written as \[ \prod_{n=1}^{\infty} (1 – q^n) = \sum_{k=-\infty}^{\infty} (-1)^k q^{k(3k-1)/2} \] Now what happens if we add \(\frac{1}{24}\) to \(\frac{k(3k-1)}{2}\)? \[ \frac{k(3k – 1)}{2} + \frac{1}{24} = \frac{12 \cdot k(3k – 1)}{24} + \frac{1}{24}=\frac{(6k-1)^2}{24} \] So the fraction \(\frac{1}{24}\) turns the exponent into a square divided by 24. Write \(Q = q^{1/24} = e^{2\pi i \tau/24}\). Then \[ \eta(q)=\sum_{k=-\infty}^{\infty} (-1)^k Q^{(6k-1)^2} \] So the mysterious prefactor is not decoration. It is the adjustment factor that turns Euler’s pentagonal exponents into perfect squares. And once the exponents are squares I recognize the shape. Jacobi theta function (one of them, because there are 4 of them)!

Here I have to be careful, because the theta function is conventionally written with a different nome than the one in my definition of \(\eta\). The standard one is \[ \theta(q)=\sum_{k=-\infty}^{\infty} q^{k^2} \] The two functions are related: sums over \(\mathbb{Z}\) of a variable raised to square exponents. And the factor 24 was needed to complete the square.

I knew that the Jacobi theta function could be used to prove that all numbers can be written as a sum of four squares (Lagrange’s four square theorem), by considering the generating function

\[\theta(q)^4 = \left( \sum_{k=-\infty}^{\infty} q^{k^2} \right)^4\]

whose coefficients count the representations of each integer as an ordered sum of four squares. Since the theta function is a powerful tool in number theory I suspected that the eta function would also be a powerful tool in number theory. And that would explain why I should care about the definition of the eta function.

Now the theta function can be used for studying sums of squares. But what can the eta function be used for?

I looked at \[ \prod(1-q^n) = 1 – q – q^2 + q^5 + q^7 – q^{12} – q^{15} + q^{22} + q^{26} – … \]

But what is the reciprocal (inverse) of that? \[ \frac{1}{\prod(1-q^n)}=(1+q+q^{1+1}+…)(1+q^2+q^{2+2}+…)… \] Each factor chooses how many copies of a given part to use: the first factor supplies copies of 1, the second copies of 2, and so on. For instance \(q^{1+1} q^2=q^4\) catches one of the ways we can write 4 as a sum of positive integers. So in general, the coefficient of \(q^N\) counts the ways of writing \(N\) as a sum of positive integers. So this is the generating function for integer partitions. I know that integer partitions are important. And there is the reason (at least one of the reasons) for why I should care about the eta function.

So I will stop here. But I’m still not inside. At least five doors remain closed—doors I do not intend to open now, but perhaps in some future essay.

First: I have found one symmetry, \(\tau \mapsto \tau+1\), and it cost me a root of unity. Is there another one?

Door 1. I know that the Jacobi theta function obeys \[ \theta(-1/ \tau) = \sqrt{-i \tau} \cdot \theta(\tau) \] Is something similar true for \(\eta\)?

Door 2. I know that theta function and its even powers are modular forms. Is something like this true for \(\eta\) as well?

Door 3. Earlier I got \(\eta(\tau+1)=e^{2 \pi i/24}\,\eta(\tau)\). That suggest I should examine the function \(\Delta( \tau) = \eta( \tau)^{24}\) instead.

Door 4. There is a physical reading of this. But for now take it for what it is: hand waving. A quantum string vibrates in modes numbered \(1, 2, 3, \ldots\) Mode \(n\) costs energy \(n\). Let’s count the states with a fixed energy N. Let’s say N=4. How many combinations are there?

Four quanta in mode 1. That is \(1+1+1+1\).

Two in mode 1, one in mode 2. That is \(1+1+2\).

Two in mode 2. That is \(2+2\).

One in mode 1, one in mode 3. That is \(1+3\).

One in mode 4. That is \(4\).

That gives us \(p(4)=5\).

We are counting a partition of \(N\).

Even the empty string has energy. Each mode is a quantum harmonic oscillator, and a quantum oscillator cannot be perfectly still. If it were — position and momentum both exactly zero — you’d have violated the uncertainty principle. So the lowest state of an oscillator of frequency \(\omega\) still carries energy \(\omega\)/2.

Adding up the floor of every mode, with mode \(n\) having frequency \(n\) gives: \[ E_0=\frac{1}{2}(1+2+3+…) \] Now the strange part is that equals sign is not an equals sign.

In fact if we introduce a regulator \(\epsilon = 1/ \Lambda\) we can write our sum as \[ E_0(\Lambda) = \frac{1}{2} \sum_{n=1}^{\infty} n e^{-n/\Lambda} = \frac{\Lambda^2}{2} – \frac{1}{24} + \mathcal{O}(1/\Lambda^2) \] The \(\Lambda ^2\) term is disregarded by the physicists because it depends on the regulator. Choose another regulator and you get something else a log or something, but -1/24 appears every time. And this is a deep question in physics because the terms that go to infinity appear at more places in physics. Sometimes they can be handled with renormalization and sometimes not. The series \(1+2+3+\cdots\) has no sum, but if we disregard the \(\Lambda ^2\) term we get \[ E_0 =- \frac{1}{24} \] The uncanny fact is that you get this disregard of the blowing up term directly from mathematics, by the analytic continuation, the object Riemann built. By using that analytic continuation the series \(1+2+3+\cdots\) sums to \(\zeta(-1)=-\frac{1}{12}\). So the vacuum energy is given by the math using analytic continuation is also

\[ E_0 = \zeta(-1) / 2 =- \frac{1}{24} \] So my prefactor is the vacuum energy, sign flipped. \[ \eta = q^{-E_0} \prod_{n=1}^{\infty} (1 – q^n) \] So there is a connection between quantum physics and number theory and it could be explored more.

(There are other such connections. Hugh Montgomery conjectured a formula for the statistical spacing between the zeros of the Riemann zeta function, working purely within number theory. Freeman Dyson, meeting him at tea in Princeton in 1972, recognised it on sight: it was the pair correlation function of the eigenvalues of random matrices, the same statistics that describe the energy levels of heavy atomic nuclei. Neither had gone looking for the other’s subject. The link existed only because one man happened to be holding both halves.)

Door 5. The first four doors have names, and that should make me suspicious. Door 1 is theta’s transformation law, Door 2 is modular forms, Door 3 is the discriminant, Door 4 is the connection to physics— all four I already knew about before I opened this paper. My exploration has a stopping rule, and that rule is the arrival of familiarity. The depth I reach is bounded by the shape of my existing knowledge rather than by the depth of the object itself. There is nothing wrong with this—in fact, it is necessary. The method is a homecoming procedure: it reliably walks the unfamiliar back to something I already know. But it is wrong to call it a complete success. Genuinely new structure—the stuff with no anchor anywhere in my existing map—either gets skipped, or gets misfiled as the nearest familiar thing.

For instance, I decompressed the 24 until it turned into “the shift that completes the square,” theta lit up, and I stopped. Not because the 24 was exhausted, but because something familiar had arrived and the arrival felt like an answer. But why stop there. Exploit the suspicions. Here are some things I suspect have connections but they do not sit on my shelves.

1. How are \((−1)^k\) and the exponents \((6k−1)^2\) connected?
2. I know that \(\frac{1}{24}=−\zeta(−1)/2\). I also know that \(\zeta(2)=\pi^2/6\). And I know that Re(s)=1/2 is special for the \(\zeta\) function. It is a kind of mirror. So \(s \mapsto (1-s)\) are connected. which mean that \(\zeta(2)\) and \(\zeta(-1)\) are connected. But how exactly?

Reflection My goal was to understand the given cold definition of the eta function. When I started to relate it to what I already know, things started to turn up. Things like:

Pentagonal Number Theorem (Combinatorics)

Modular Forms (Number Theory)

SL(2, Z) (Group Theory)

Theta Functions & Jacobi Identity (Analysis)

Bosons and Vacuum Energy (Physics)

That web of interconnected ideas isn’t a distraction—it is the actual architecture of high-level mathematics. The Dedekind’s eta function, acts as a grand central station for half a dozen major fields. Each branch is a rabbit hole. And if I start to crawl down one rabbit hole, it will lead to new rabbit holes. It’ easy to spend a lifetime crawling those rabbit holes.

When every node connects to five other dense subjects, learning can quickly feel like trying to map an infinite graph. How do I avoid getting lost in this hyperspace? One of my favorite methods is to trace the historical path which often mirrors intuitive connections. Euler found the pentagonal number identity purely combinatorially; Jacobi tied it to elliptic functions; Dedekind recognized its modular properties; physicists much later found it in string worldsheets. Grounding the rabbit holes in their historical order provides a natural linear narrative.

I started this essay by dissecting the definition of the eta function. What marked this definition as load-bearing was that it had an unexplained constant in it. Arbitrary-looking constants are almost always compressed theorems.

I won’t always have the afternoon to dissect a definition, not even a load bearing one. But even a partial dissection gets my foot in the door. And that opening is often worth more than the map – It is what leads to mastery.