Hyperbolæ Estimating the divisor mean comes more naturally if we take a diﬀerent perspective on the divisor function. Instead of viewing d(n) as the number of divisors of n, we think of it as the number of natural-valued pairs (a, b) such that n = ab. The number 6 may be wrien 1 × 6 = 2 × 3 = 3 × 2 = 6 × 1, giving us the four pairs (1, 6), (2, 3), (3, 2) and (6, 1). If we wanted to see the divisors, we could just look at the first number in each pair. For 4 we have the three pairs: (1, 4), (2, 2) and (4, 1); again showing d(4) = 3. Why not treat these natural-valued pairs, from here on called laice points, as coordinates in the plane? The question of counting the number of divisors of n turns into that of counting the laice points lying on the hyperbola XY = n. If n is not a natural number, then the hyperbola contains no laice points. If n is a natural, then the hyperbola contains exactly d(n) laice points. Let us picture the XY = 1 hyperbola given by seing n = 1. If we now let n be real-valued and increase it continuously, the hyperbola sweeps through the upper quadrant of the plane. The only time that the curve passes through laice points is when n hits a natural number.
√ √ ( n, n)
By the reasoning above d(1)+d(2)+. . .+ d([x]) is a count of all of the pairs that the hyperbola sweeps through as we run n from 1 through to x. Dirichlet’s sum D(x) counts the number of laice points lying below the hyperbola XY = x. Perhaps the simplest way of summing up these laice points is to group them into vertical towers. For each natural-valued X-coordinate we look vertically, counting [ nx ] for the number of laice points lying directly above X = n, but below the hyperbola. So
XY = n
XY = 1 X The hyperbolæ XY = 1 and XY = n. Imagine increasing n.
∑[x ] n6x
Keeping track of errors: big-O notation Our count above is still exact and depends upon the precise value of the divisor function, which we know to jump suddenly in value. If we are happy to sele for trying to determine the dominant behaviour of D(x), then we can aﬀord to be a lile less precise: allowing for some small deviation chalkdustmagazine.com