# Chapter ᚑ(O): The prime mileposts
*Teclam Áirme, the Mathematical Compendium — a copy for AI readers of https://www.teclamairme.com/#ch-T18, as of 2026-09-30. Links are relative to https://www.teclamairme.com/fortheaireaders/.*
[Guide for AI readers](index.md) · [Its glossary cards](ch18-cards.md) · [Its references](ch18-refs.md) · [Previous chapter](ch17.md) · [Next chapter](ch19.md)
*A footnote such as [r-18.3] is reference entry 18.3 and links to it. Section 18 of 28; chapters are named by their Ogham letter, never by number.*
ᚑ – [Onn](ch18-cards.md#g-og-onn): associated with the gorse, Section 18
## Chapter ᚑ(O): The prime mileposts
> “It is very probable that all roots are real. Certainly one would wish for a stricter proof here; I have meanwhile temporarily put aside the search for this after some fleeting futile attempts, as it appears unnecessary for the next objective of my investigation.”
> — Bernhard Riemann · On the Number of Prime Numbers less than a Given Quantity, 1859, in David Wilkins’s translation (the roots are those of Riemann’s transformed function; this is the critical-line claim) [r-18.25](ch18-refs.md#r-18.25)
Formula: ([ζ(−2)](ch18-cards.md#g-zetam2) + [ζ(½ + i·t1)](ch18-cards.md#g-zetarho) + [1/ζ(1)](ch18-cards.md#g-recipzeta1) + [ζ(2)∏p prime(1−p−2)−1](ch18-cards.md#g-eulerproduct2))/(([ζ(2)](ch18-cards.md#g-zeta2)·[ζ(3)](ch05-cards.md#g-apery)·[ζ(4)](ch18-cards.md#g-zeta4)·[ζ(6)](ch18-cards.md#g-zeta6)·|[ζ(0)](ch18-cards.md#g-zeta0)|·|[ζ(−1)](ch18-cards.md#g-zetam1)|·[|ζ(½)|](ch18-cards.md#g-zetahalf)) {brace: values} · ([t1](ch18-cards.md#g-t1)·[t2](ch18-cards.md#g-t2)) {brace: [zeros](ch28-cards.md#g-critline)} · ([(1 − 2−2)−1](ch18-cards.md#g-eulerfactor2)) {brace: [the Euler product](ch18-cards.md#g-eulerprod)} ) {brace: [analytic number theory](ch18.md#ch-T18-analytic-number-theory)}
#### Type of number
Proven transcendental · Status open ·
Key concepts: Analytic number theory: the Riemann zeta function and the Euler product. It includes symbols you may not have seen before, such as: [ζ(s)](ch18-cards.md#g-zeta), [∏](ch18-cards.md#g-eulerprod), and [t1](ch18-cards.md#g-t1). Zero here comes from zeta four ways: a tame zero at −2, a wild one on the critical line, the pole at 1 turned upside down, and Euler’s product canceling zeta exactly.
The mopeds are zippier now! Let me call out the prime-numbered mileposts as we pass; you’ll soon notice that they do not come at regular intervals. The workshop left us a way to estimate how many we should meet. [Zeta](ch18-cards.md#g-zeta) gives us another way to read the road, revealing the primes inside a sum that seems to concern every positive integer.
Field notes
### [analytic number theory](ch18.md#ch-T18-analytic-number-theory) {#ch-T18-analytic-number-theory}
Begin with [ζ(s)](ch18-cards.md#g-zeta) = 1 + 1/2ˢ + 1/3ˢ + ⋯. At s = 2, Euler found [π²/6](ch18-cards.md#g-zeta2): the circle appears in a sum over squares. More surprisingly, the same sum can be rebuilt by multiplying contributions from individual primes. [r-18.2](ch18-refs.md#r-18.2) [r-18.4](ch18-refs.md#r-18.4)
The series adds up to a finite total only when s is bigger than 1; s may also be a complex number, and then the test is that its real part, Re(s), be bigger than 1. Everywhere else, zeta is defined by [analytic continuation](ch18-cards.md#g-analcont): there is exactly one well-behaved extension that agrees with the series where it converges, and it breaks down at just one point, its pole at s = 1. That is how zeta gets values the series cannot give. Put s = −1 into the series and it reads 1 + 2 + 3 + ⋯, which diverges; the continued function still has a value there, [ζ(−1)](ch18-cards.md#g-zetam1) = −1/12. Continuation and summing are different operations. [r-18.1](ch18-refs.md#r-18.1)
#### Numerator
[ζ(−2)](ch18-cards.md#g-zetam2) = 0; the first of zeta’s *trivial zeros*, the ones we can explain.
The [functional equation](ch18-cards.md#g-funceq) relating [ζ(s)](ch18-cards.md#g-zeta) to ζ(1 − s) contains a sine factor that vanishes at every negative even integer, while the other factors stay finite. “Trivial” marks these zeros as explained, not as easy to discover. [r-18.5](ch18-refs.md#r-18.5)
[ζ(½ + i·t1)](ch18-cards.md#g-zetarho) = 0; the first *nontrivial* zero, the kind [the Riemann Hypothesis](ch18-cards.md#g-rh) is about.
The inputs ½ + i·t run straight up a vertical line in the complex plane, the [critical line](ch28-cards.md#g-critline) Re(s) = ½, and t says how far up: its height. [t₁](ch18-cards.md#g-t1) ≈ 14.1347 is the height of the first zero on that line, first published by Gram more than a century ago. The term uses the exact height, defined by the zero itself, so it is exactly zero; plug in the rounded 14.1347 and you would get a small nonzero value. [r-18.7](ch18-refs.md#r-18.7)
[1/ζ(1)](ch18-cards.md#g-recipzeta1) = 0; zeta’s one pole, turned upside down, as [the gamma function](ch17-cards.md#g-gammafn)’s was in Chapter ᚐ(A).
At s = 1 the series becomes the harmonic series 1 + ½ + ⅓ + ⋯, which grows without limit, and the continued zeta keeps that as its one pole: near s = 1 it behaves like 1/(s − 1), shooting off to infinity. Turn it upside down and the reciprocal behaves like s − 1, which passes smoothly through zero at s = 1. [r-18.6](ch18-refs.md#r-18.6)
[ζ(2)∏p prime(1−p−2)−1](ch18-cards.md#g-eulerproduct2) = 0; the [Euler product](ch18-cards.md#g-eulerprod), the reason a sum over all integers knows about primes, turned into a zero.
Expand each (1 − p⁻ˢ)⁻¹ as 1 + p⁻ˢ + p⁻²ˢ + ⋯, so each prime milepost supplies all its powers. Multiply over every prime, and [unique factorization](ch18-cards.md#g-ufd) makes every positive integer appear exactly once: 12⁻ˢ, for instance, turns up only as 2⁻²ˢ from the prime 2’s factor times 3⁻ˢ from the prime 3’s, because 12 = 2² × 3 and in no other way. So [ζ(s)](ch18-cards.md#g-zeta) = ∏p prime(1 − p−s)−1 for Re(s) > 1. At s = 2, multiplying [ζ(2)](ch18-cards.md#g-zeta2) by the product of the factors (1 − p⁻²) undoes the [Euler product](ch18-cards.md#g-eulerprod) exactly, so the result is 1, and subtracting 1 leaves zero. [r-18.4](ch18-refs.md#r-18.4) [r-18.16](ch18-refs.md#r-18.16)
#### Denominator
Three groups, multiplied: values of zeta, the places where it vanishes, and one prime’s contribution to the [Euler product](ch18-cards.md#g-eulerprod).
values
[ζ(2)](ch18-cards.md#g-zeta2) = π²/6; the Basel value, Euler’s 1734 triumph.
The sum of the reciprocal squares; the even positive inputs all give Euler’s circle formulas. [r-18.2](ch18-refs.md#r-18.2) [r-18.3](ch18-refs.md#r-18.3)
[ζ(3)](ch05-cards.md#g-apery) ≈ 1.2021; Apéry’s constant, back from Chapter ᚃ(F), the odd value whose transcendence is unknown.
[The sum of the reciprocal cubes](ch05-cards.md#g-apery), 1 + 1/8 + 1/27 + ⋯: every term is positive, so the sum is too. Apéry proved it irrational in 1978. [r-18.9](ch18-refs.md#r-18.9)
[ζ(4)](ch18-cards.md#g-zeta4), [ζ(6)](ch18-cards.md#g-zeta6) = π⁴/90, π⁶/945; two more of Euler’s even values.
Positive sums of reciprocal powers, each a rational multiple of a power of π. [r-18.2](ch18-refs.md#r-18.2)
|[ζ(0)](ch18-cards.md#g-zeta0)|, |[ζ(−1)](ch18-cards.md#g-zetam1)| = ½, 1/12; the scandalous values, from continuation rather than summing.
Continuation gives [ζ(0)](ch18-cards.md#g-zeta0) = −½ and [ζ(−1)](ch18-cards.md#g-zetam1) = −1/12, both negative; the absolute-value bars turn them into the positive ½ and 1/12. [r-18.2](ch18-refs.md#r-18.2) [r-18.18](ch18-refs.md#r-18.18)
[|ζ(½)|](ch18-cards.md#g-zetahalf) ≈ 1.4604; zeta at s = ½, the one real point on the [critical line](ch28-cards.md#g-critline).
[ζ(½)](ch18-cards.md#g-zetahalf) ≈ −1.4603545…, so its absolute value is positive. No source cited here settles whether it is irrational. [r-18.10](ch18-refs.md#r-18.10)
zeros
[t1](ch18-cards.md#g-t1) · [t2](ch18-cards.md#g-t2) ≈ 14.1347 × 21.0220; the heights of the first two nontrivial zeros.
These are *locations* of zeros, positive heights, not zero function values, so a chapter about zeros can safely put them underneath a fraction. Both lie on the [critical line](ch28-cards.md#g-critline). [r-18.7](ch18-refs.md#r-18.7)
the Euler product
[(1 − 2−2)−1](ch18-cards.md#g-eulerfactor2) = 4/3; the prime 2’s own share of [ζ(2)](ch18-cards.md#g-zeta2).
(1 − ¼)⁻¹ = 4/3: one positive piece of the [Euler product](ch18-cards.md#g-eulerprod). Multiply the factors for every prime and you rebuild [π²/6](ch18-cards.md#g-zeta2). [r-18.16](ch18-refs.md#r-18.16)
The only fraction is 0/(values · zeros · the [Euler product](ch18-cards.md#g-eulerprod)): every factor below the line is finite and positive, so the denominator is too, for reasons that do not depend on [the Riemann Hypothesis](ch18-cards.md#g-rh).
**Fine print.** All nontrivial zeros lie in [the critical strip 0 < Re(s) < 1](ch18-cards.md#g-critstrip). [The Riemann Hypothesis](ch18-cards.md#g-rh) asks whether they all lie on its center line Re(s) = ½; our two selected zeros do, which settles nothing for the rest. This is where the mileposts come back. Riemann’s explicit formula counts the primes exactly: a smooth main term built from the [logarithmic integral](ch17-cards.md#g-li), the tool we brought from the workshop in Chapter ᚐ(A), plus a wobble from every nontrivial zero and a few small terms besides. The zeros are why the [prime count](ch18-cards.md#g-primecount) drifts above and below the smooth estimate the [prime number theorem](ch18-cards.md#g-pnt) supplies, and the hypothesis would hold that drift to the tightest bound possible. [r-18.12](ch18-refs.md#r-18.12) [r-18.8](ch18-refs.md#r-18.8) [r-18.19](ch18-refs.md#r-18.19) [r-18.28](ch18-refs.md#r-18.28) [1/ζ(1)](ch18-cards.md#g-recipzeta1) means the extended reciprocal at the pole; ζ(1) is not given a finite value. The final −1 in the fourth numerator term is subtraction, not one of the product’s inverse powers.
What this chapter adds to the formula: 0 / (values · zeros · [the Euler product](ch18-cards.md#g-eulerprod)) = 0
Zeta has revealed the primes inside a sum over all positive integers. Its established identities give this chapter zero, while the question about its nontrivial zeros waits for the homeward view, when the whole route lies beneath the window.
Ease off a little; there’s a crossroads ahead. Pull over by the signpost and we can get the map out. The primes have brought us to a place where lines and curves meet, and I want to show you a rather unexpected way of adding.
Watch and read
- [But what is the Riemann zeta function? Visualizing analytic continuation](https://www.youtube.com/watch?v=sD0NjbwqlYw) 3Blue1Brown, 2016: the best picture of what continuation actually does to the plane
- [The Riemann Hypothesis, Explained](https://www.youtube.com/watch?v=zlm1aajH6gY) Quanta Magazine with Alex Kontorovich: the primes, the zeros, and why the prize exists
- [Riemann zeta function](https://en.wikipedia.org/wiki/Riemann_zeta_function) Wikipedia
- [Riemann hypothesis](https://en.wikipedia.org/wiki/Riemann_hypothesis) Wikipedia