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Concept

Error term

The difference between a quantity and the approximation used for it, usually written as a bound on its size. Much of analytic number theory consists of shrinking error terms, where the main term is easy and the error decides everything.

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

The lattice-point count inside a circle, less its area, out to radius 160. A plot of the difference between the number of lattice points in a disc and the disc's area, against radius, with envelopes proportional to the square root and the two-thirds power drawn.

The dots a circle catches

Pick's theorem gives a lattice polygon's area exactly, with no error term anywhere. Ask a circle the same question and the exactness is gone: the count is the area plus something, the something has been measured for two centuries, and nobody knows how big it is.

discrete · Pick theorem The sum of the first 6 squares, as a staircase over a curve. Bars of height k^2 for k from 1 to 6, totalling 91, drawn over the curve y = x^2, whose area up to 6 is 72.00. The slivers between staircase and curve hold 19.00, close to half the last bar.

Sums of powers, read off a staircase

Add the first n squares, or cubes, or seventh powers, and the answer is always a polynomial in n. Its first term is the area under a curve, its second is half of the last step, and every term after that is a correction for the corners of a staircase — which is where the Bernoulli numbers come from, and why they eventually grow without bound.

geometry · Figurate numbers

Named alongside it

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

ApproximationAsymptotic seriesAsymptoticsDivisor functionExhaustive searchFigurate numbersIntegralLatticeOpen problemPick theoremPolynomialRiemann sum

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