Concept

Irrationality measure

The exponent saying how closely a number can be approached by fractions relative to their denominators. Rational numbers have measure one and every irrational number at least two; almost every number has exactly two, and a proof of irrationality usually bounds the measure from above too.

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

The Flint Hills series, summed to ten million terms. A staircase of partial sums on a logarithmic scale of n, jumping at 1, 3, 22 and 355 and flat afterwards near 30.3.

A series that waits on π

Add 1/(n³ sin² n) for n = 1, 2, 3, … and the terms are mostly tiny, except where n is almost a multiple of π and sin n is almost nought. Ten million terms add to 30.3145, four-fifths of it from the single term at n = 355. Whether the sum is finite depends on how closely fractions can approach π — on a number called its irrationality measure — and the best proof available says only that the measure is below 7.1, where the series needs it below 2.5.

analysis · The integral

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

Named alongside it

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

Continued fractions Convergence Diophantine approximation Irrationality Least common multiple Partial sum Pi Prime number theorem Recurrence relation Riemann zeta function Series Small divisors

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