Reference

Glossary

Every technical term on the site — defined briefly, honestly, and at the right resolution. Dotted-underlined words in any chapter link here.

Prime numberLevel 1 →: A whole number greater than 1 divisible only by 1 and itself: 2, 3, 5, 7, 11, … The multiplicative atoms of the integers.
Composite numberLevel 1 →: A whole number greater than 1 that is not prime — i.e. it factors as a product of smaller numbers, hence (by the fundamental theorem) of primes.
Fundamental theorem of arithmeticLevel 1 →: Every integer greater than 1 is a product of primes in exactly one way, up to the order of the factors. Proved in essence in Euclid’s Elements; the bedrock under everything on this site, including Euler’s product.
Sieve of EratosthenesLevel 1 →: The ancient algorithm for finding primes: repeatedly keep the smallest surviving number and cross out its multiples. Still the conceptual ancestor of modern sieve methods in number theory.
ProofLevel 1 →: A logical argument establishing a statement for allcases at once — as opposed to verification, which checks finitely many. Euclid’s infinitude of primes covers infinitely many cases in five sentences; no computation could.
Ulam spiralLevel 1 →: Integers written in a square spiral with primes highlighted (Stanisław Ulam, 1963). Primes visibly cluster on certain diagonals — the values of certain quadratic polynomials — hinting at structure no one has fully explained.
Prime-counting function π(x)Level 2 →: The number of primes less than or equal to x. Examples: π(10) = 4, π(100) = 25, π(10⁶) = 78,498. The protagonist of Level 2.
Natural logarithm (ln)Level 2 →: The inverse of the exponential function: ln(x) is the time for continuous 100% growth to reach x from 1. Turns multiplication into addition and grows glacially — ln(10⁶) ≈ 13.8. The density of primes near x is about 1/ln(x).
Logarithmic integral Li(x)Level 2 →: $\operatorname{Li}(x) = \int_2^x dt/\ln t$ — Gauss’s refined estimate for π(x), obtained by adding up the local density 1/ln t. Under RH it is accurate to roughly square-root precision.
Prime Number Theorem (PNT)Level 2 →: $\pi(x) \sim x/\ln x$ : the ratio of the two sides tends to 1. Conjectured by Gauss and Legendre, proved in 1896 by Hadamard and de la Vallée Poussin via the zeros of ζ.
Skewes' numberLevel 2 →: A (gigantic) bound on the first x where π(x) overtakes Li(x). Littlewood proved the lead changes infinitely often, yet no explicit crossing is known — the standard parable for why “verified very far” proves nothing in number theory.
Harmonic seriesLevel 3 →: 1 + ½ + ⅓ + ¼ + … — diverges to infinity, with extreme slowness (about 10⁴³ terms to pass 100). Its divergence becomes the pole of ζ(s) at s = 1.
Riemann zeta function ζ(s)Level 3 →: For Re(s) > 1, $\zeta(s) = \sum_{n\ge1} n^{-s}$ ; elsewhere, its unique analytic continuation (one pole, at s = 1). Knows the primes through Euler’s product.
Basel problemLevel 3 →: Find the exact value of 1 + ¼ + ⅑ + 1/16 + … Euler’s 1735 answer, $\pi^2/6$ , made his name and revealed that ζ hides deep structure.
Euler product (the golden key)Level 3 →: $\sum_n n^{-s} = \prod_p (1 - p^{-s})^{-1}$ for Re(s) > 1 — one factor per prime, valid because factorization is unique. The bridge between analysis and the primes.
Complex numberLevel 3 →: A number a + bi with i² = −1; equivalently a point (a, b) in the plane. Addition is arrow-addition; multiplication rotates and stretches. Re(s) and Im(s) denote the two coordinates.
Analytic functionLevel 3 →: A complex function expressible locally as a convergent power series. Analytic functions are extraordinarily rigid: their values on any small patch determine them everywhere they exist. This rigidity powers analytic continuation.
Analytic continuationLevel 3 →: The (unique, when it exists) extension of an analytic function beyond the domain of its defining formula — as 1/(1−x) extends 1 + x + x² + … beyond |x| < 1. Riemann continued ζ to the whole plane minus s = 1.
PoleLevel 3 →: A point where an otherwise analytic function blows up to infinity in a controlled way. ζ has exactly one, at s = 1 — the ghost of the divergent harmonic series.
Zero (of a function)Level 3 →: An input where the function’s output is 0. In domain coloring, zeros of ζ appear as black wells where all hues meet.
Trivial zerosLevel 3 →: The zeros of ζ at s = −2, −4, −6, …, manufactured by the factor sin(πs/2) in the functional equation. Fully understood, hence “trivial.”
Nontrivial zerosLevel 4 →: All other zeros of ζ. They lie in the critical strip, symmetric about both the real axis and the critical line; the first is at ½ + 14.134725…i. The explicit formula turns each into a wave through the primes. RH says they all have real part ½.
Critical stripLevel 3 →: The vertical band 0 < Re(s) < 1 of the complex plane — the only region where ζ’s zeros are not fully understood, and home of all nontrivial zeros.
Critical lineLevel 4 →: The line Re(s) = ½ down the middle of the critical strip — the mirror axis of the functional equation. RH: every nontrivial zero sits on it.
Riemann Hypothesis (RH)Level 4 →: Every nontrivial zero of ζ has real part ½ (Riemann, 1859). Equivalent to the prime-counting error being square-root small: |π(x) − Li(x)| = O(√x ln x). Unproven; a Clay Millennium Prize problem.
Explicit formulaLevel 4 →: Riemann’s exact identity rebuilding the prime staircase from the zeros: $\psi(x) = x - \sum_\rho x^\rho/\rho - \ln 2\pi - \tfrac12\ln(1-x^{-2})$ . Each zero contributes one oscillation — the “music of the primes.”
Functional equationLevel 3 →: $\zeta(s) = 2^s \pi^{s-1} \sin(\pi s/2)\,\Gamma(1-s)\,\zeta(1-s)$ — the reflection law tying s to 1 − s, with the critical line as mirror. It creates the trivial zeros and forces nontrivial zeros into symmetric pairs.
GUE (Gaussian Unitary Ensemble)Level 5 →: A standard random-matrix model from quantum physics. Its eigenvalue statistics — including level repulsion — match the statistics of zeta zeros with uncanny precision (Montgomery–Odlyzko phenomenon).
Hilbert–Pólya conjectureLevel 5 →: The hope that the nontrivial zeros are the spectrum of some self-adjoint operator — which would make their reality (hence RH) automatic, the way quantum energy levels are automatically real. Unrealized, but the random-matrix evidence points its way.
Generalized Riemann Hypothesis (GRH)Level 5 →: RH extended to the whole family of Dirichlet L-functions (and beyond): all their nontrivial zeros on their critical lines. Governs primes in arithmetic progressions; many algorithmic results are conditional on it.
de Bruijn–Newman constant ΛLevel 5 →: A real parameter measuring how much one can “heat-flow” ζ before zeros could leave the line; RH is the statement Λ ≤ 0. Rodgers–Tao (2018) proved Λ ≥ 0: if RH holds, it holds with nothing to spare.
Millennium Prize ProblemsLevel 5 →: Seven problems selected by the Clay Mathematics Institute in 2000, each carrying a $1,000,000 prize. RH is one; only the Poincaré conjecture has been solved (Perelman, who declined the prize).