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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 3 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 A ball counted slice by slice. Ball of radius 7.5: slices 0:177, 1:177, 2:169, 3:145, 4:129, 5:97, 6:69, 7:21; total 1791, volume 1767.146.

The dots a ball catches

Count the lattice points inside a ball and compare with its volume. In five dimensions the error is about the size of one spherical shell, and that is a theorem; in four a logarithm sneaks in; in three and two the true size of the error is unknown. The difficulty runs backwards because the arithmetic of sums of squares is most erratic when there are fewest squares to add.

discrete · Pick theorem

Named alongside it

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

AsymptoticsDivisor functionApproximationAsymptotic seriesClass numberDimensionExhaustive searchFigurate numbersIntegralLatticeLattice pointOpen problem

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