From the Reflection Trick to Fermi Integrals and Apéry's Constant

2026-04-21

The previous post solved an MIT Integration Bee problem with a two-line collapse built around the identity

$$\frac{1}{e^x+1} + \frac{1}{e^{-x}+1} = 1.$$

That trick lives on a bounded symmetric interval $[-a, a]$. It pairs the logistic factor with its reflection to eliminate it.

But the same logistic factor has a second life on the half-line $[0, \infty)$, where reflection is no longer available but the logistic factor still has deep structure. That half-line life connects directly to the Riemann zeta function, to the Dirichlet eta function, and to Apéry’s constant $\zeta(3)$ — the subject of one of twentieth-century number theory’s surprise theorems.

This post walks from one life to the other.

1. A Different Evaluation, Same Factor

Consider the integral

$$F(s) := \int_0^{\infty} \frac{x^{s-1}}{e^x+1}\, dx, \qquad s > 0.$$

This is a Fermi integral (the name comes from statistical mechanics: $1/(e^x+1)$ is the Fermi–Dirac distribution, and integrals against energy weights compute thermodynamic averages at fermionic populations).

The reflection trick does not apply here — the domain is one-sided, not symmetric — but the logistic factor still decomposes cleanly, this time via a geometric series.

Rewrite:

$$\frac{1}{e^x+1} = \frac{e^{-x}}{1+e^{-x}}.$$

For $x > 0$, the quantity $e^{-x}$ is in $(0, 1)$, so $1/(1+e^{-x})$ is the sum of a geometric series with ratio $-e^{-x}$:

$$\frac{1}{1+e^{-x}} = \sum_{k=0}^{\infty} (-1)^k e^{-kx}.$$

Multiplying by the $e^{-x}$ out front and re-indexing ($n = k+1$):

$$\frac{1}{e^x+1} = \sum_{n=1}^{\infty} (-1)^{n-1} e^{-nx}.$$

The logistic factor is therefore an alternating sum of decaying exponentials. This is a very different description from the complementary-pair identity of the bounded-interval post — but it is the description that makes the half-line integral tractable.

2. Evaluating the Fermi Integral

Substitute the series into $F(s)$ and exchange sum and integral (uniform convergence on $[\varepsilon, \infty)$ plus integrability at $0$ justifies this; details omitted at the first-undergraduate level):

$$F(s) = \int_0^{\infty} x^{s-1} \sum_{n=1}^{\infty} (-1)^{n-1} e^{-nx}\, dx = \sum_{n=1}^{\infty} (-1)^{n-1} \int_0^{\infty} x^{s-1} e^{-nx}\, dx.$$

Each inner integral is a standard Gamma function computation. Substitute $u = nx$:

$$\int_0^{\infty} x^{s-1} e^{-nx}\, dx = \frac{1}{n^s}\int_0^{\infty} u^{s-1} e^{-u}\, du = \frac{\Gamma(s)}{n^s}.$$

Therefore

$$F(s) = \Gamma(s) \sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n^s} = \Gamma(s)\,\eta(s),$$

where

$$\eta(s) := \sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n^s}$$

is the Dirichlet eta function.

So: the Fermi integral $F(s)$ is simply $\Gamma(s)$ times the Dirichlet eta function. Two famous objects, one integral.

3. From Eta to Zeta

The eta function is a close relative of the Riemann zeta function

$$\zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s},$$

differing only in the sign pattern. The relationship between them is elementary and worth doing on paper once:

$$\zeta(s) - \eta(s) = \sum_{n=1}^{\infty} \frac{1 - (-1)^{n-1}}{n^s} = \sum_{n \text{ even}} \frac{2}{n^s} = 2 \sum_{m=1}^{\infty} \frac{1}{(2m)^s} = 2 \cdot 2^{-s}\, \zeta(s) = 2^{1-s}\,\zeta(s).$$

Solving for $\eta(s)$:

$$\boxed{\eta(s) = \bigl(1 - 2^{1-s}\bigr)\,\zeta(s).}$$

Combining with the previous section:

$$\int_0^{\infty} \frac{x^{s-1}}{e^x+1}\, dx = \Gamma(s)\bigl(1 - 2^{1-s}\bigr)\,\zeta(s).$$

Every Fermi integral is, up to elementary factors, a value of the Riemann zeta function.

4. The Apéry Constant

Specializing to $s = 3$:

$\Gamma(3) = 2! = 2.$
$1 - 2^{1-3} = 1 - 1/4 = 3/4.$

Therefore

$$\int_0^{\infty} \frac{x^{2}}{e^x+1}\, dx = 2 \cdot \frac{3}{4}\,\zeta(3) = \frac{3}{2}\,\zeta(3).$$

The number $\zeta(3) \approx 1.20205690\ldots$ is known as Apéry’s constant.

Until 1978 it was conjectured, but not known, whether $\zeta(3)$ is irrational. Roger Apéry announced an elementary proof at a conference in Marseille in June 1978, using a startling accelerated series:

$$\zeta(3) = \frac{5}{2}\sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n^3 \binom{2n}{n}}.$$

The audience was skeptical at first — Apéry’s presentation was terse and his methods unconventional — but the proof was verified within weeks, and $\zeta(3)$ was confirmed irrational. Whether $\zeta(3)$ is transcendental remains open to this day. Whether $\zeta(5), \zeta(7), \ldots$ are irrational is also still open. (It is known, by a theorem of Ball and Rivoal, that infinitely many of the odd zeta values are irrational, but no specific one beyond $\zeta(3)$ has been pinned down.)

5. The Full Picture

Two integrals, both built around the logistic factor $1/(e^x+1)$, bracket the terrain:

$$\int_{-1}^{1} \frac{dx}{(e^x+1)(x^2+1)} = \frac{\pi}{4}, \qquad \int_{0}^{\infty} \frac{x^2}{e^x+1}\, dx = \frac{3}{2}\zeta(3).$$

The left integral is a competition problem whose answer is a piece of elementary geometry ($\arctan$-related) and whose evaluation takes two lines using the reflection identity.

The right integral is a classical object of analytic number theory whose answer is Apéry’s constant, and whose evaluation takes a geometric-series expansion followed by a Gamma computation.

Same factor, two different techniques, two different worlds of mathematics:

Domain	Technique	Structure exploited	Answer
$[-1,1]$, symmetric	Reflection: $x \mapsto -x$	$\dfrac{1}{e^x+1} + \dfrac{1}{e^{-x}+1} = 1$	$\pi/4$
$[0,\infty)$, half-line	Series: $\sum (-1)^{n-1} e^{-nx}$	Gamma integral + $\eta(s) = (1-2^{1-s})\zeta(s)$	$\frac{3}{2}\zeta(3)$

The logistic function is one of the most versatile objects in analysis: a sigmoid in machine learning, the Fermi distribution in statistical mechanics, and — via the two integrals above — a bridge between the arctangent and Apéry’s constant.

6. Why This Matters Beyond the Pretty Identity

The Fermi-integral evaluation is more than a textbook exercise. It is the analytic underpinning of several concrete things:

Thermodynamics of Fermi gases. The chemical potential, heat capacity, and pressure of a free-electron gas at temperature $T$ are all proportional to Fermi integrals $F(s)$ for integer $s$. The constant $\zeta(3)$ thus appears in the thermodynamic expansion coefficients of metals — a number-theoretic object showing up in condensed-matter physics.
Formalization of $\zeta(3)$ as a computable constant. Apéry’s constant is one of the canonical targets for modern formal-verification work in analog-computing and CRN theory. The irrationality proof (and its quantitative refinements, such as effective Diophantine approximation bounds for $\zeta(3)$) is central to the question: which real numbers can be computed by a chemical reaction network in real time, and with what resources? The Fermi-integral representation is one of several entry points to $\zeta(3)$, and it is the one most amenable to CRN encoding via integral approximation.
A template for $\zeta(2k+1)$. The same geometric-series technique handles every odd integer $s \geq 3$. $\zeta(5), \zeta(7), \zeta(9), \ldots$ all arise as Fermi-integral specializations. None of their irrationality (with the single exception of $\zeta(3)$) is known — they are open problems of central importance in transcendence theory.

One-Line Summary

The logistic factor $1/(e^x+1)$ is the same object in two mathematical worlds: on a bounded symmetric interval it is a reflection partner; on the half-line it is an entry point to the Riemann zeta function.

One factor. Two techniques. One unifying function.

溫故而知新，可以為師矣。

Reviewing the old and knowing it anew — one may thereby be a teacher.

— 《論語·為政》 · Analects, II.11