Concept

Riemann zeta function

The function ζ(s) given by the sum of 1/n^s over the positive integers, and its continuation to the complex plane. It encodes the primes through Euler's product, and its values at whole numbers, such as ζ(2) = π²/6, appear far from number theory.

Named by 2 essays across 2 fields — each of them below, with the objects they name alongside it.

Apéry's proof for ζ(3), as three numbers. log₁₀ of 2·lcm(1, …, n)³, of |a(n)ζ(3) − b(n)| and of their product for n up to 60: slopes 1.262, -1.546, -0.284 per step.

The race that makes ζ(3) irrational

Roger Apéry's 1978 proof that the sum of the reciprocal cubes is not a fraction comes down to a race between two numbers. A whole-number multiplier grows by a factor of ten every 0.79 steps; the gap it multiplies shrinks by a factor of ten every 0.65. The gap wins, by a margin of about seventeen per cent, and that margin is the whole proof.

number · Irrationality

The cheapest tree through forty points. n=40, seed 8: total 1.063675, largest edge 0.0605, ζ(3)=1.2020569.

The cheapest tree costs ζ(3)

Join n towns by every possible road, give each road a cost drawn at random between nought and one, and find the cheapest network of n − 1 roads that connects them all. However many towns there are, the bill tends to 1 + 1/8 + 1/27 + 1/64 + … = 1.2020569. The proof watches the cheap roads arrive one at a time and turns the cost into the area under a curve that counts the pieces of a growing random graph.

applied · Assignment

Named alongside it

The objects these essays reach for when they reach for this one.

Cayleys formula Central limit theorem Expectation Greedy algorithm Irrationality Irrationality measure Least common multiple Prime number theorem Random graph Recurrence relation Spanning tree

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