Concept

Mobius function

The function on whole numbers that is 0 at any number with a repeated prime factor, and otherwise +1 or −1 as the number of prime factors is even or odd. It inverts summing over divisors, and its values sum to the counts that inclusion and exclusion produce.

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

$The 46 Farey fractions of order 12, and how far each strays from even spacing. Farey fractions of order 12 against 46 evenly spaced points, with the deviation of each drawn as a bar; largest deviation 0.0616.$

How evenly the fractions spread

List every fraction between nought and one with denominator at most n, in order. They spread across the interval almost evenly, and how fast the unevenness shrinks as n grows is — exactly, provably — the Riemann hypothesis. The link runs through a second fact: set the fractions round a circle and add them as arrows, and what is left is a whole number.

number · Stern brocot

The primitive 15th roots of unity, raised to a power and added. c₁₅(1) = 1; c₁₅(3) = −2; c₁₅(5) = −4; c₁₅(6) = −2; c₁₅(10) = −4; c₁₅(15) = 8.

The sums of primitive roots are always whole

Take the roots of unity of order exactly q, raise each to the power n and add them. The answer is always a whole number, it depends on n only through what n shares with q, and as n varies the sums behave like the sines and cosines of a Fourier series — so well that Ramanujan could rebuild the sum of the divisors of any number from them, and prove that their weighted total is nought, a fact that at n = 1 is the prime number theorem.

algebra · Roots of unity

Named alongside it

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

Prime number theorem Totient Cyclotomic polynomial Discrepancy Equidistribution Farey sequence Fourier series Greatest common divisor Mertens function Riemann hypothesis Roots of unity

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