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Mean values The trouble is, the divisor function is not at all wellbehaved. In fact, it jumps significantly in value infinitely oen. The adjacent integers d(200560490130) = 211 and d(200560490131) = 2 provide one such example, but you can happily construct as large a jump as you fancy. Try it.

The divisor function is not at all well-behaved, jumping significantly in value infinitely oen.

The divisor function is one of many interesting arithmetic functions; understanding their asymptotic behaviour as n → ∞ is a key question in multiplicative number theory. However, as we saw above, the value of these functions can be erratic. To try to smooth this wildness, you could instead consider certain types of ‘average’. Dirichlet’s sum was one such average. He began by accumulating a sum from the divisor function on adjacent naturals: D(x) = d(1) + d(2) + . . . + d([x]) =

∑

d(n),

(1)

n6x

where [x] indicates the largest integer less than or equal to x. Upon dividing the sum D(x) by x, we get the mean value of the divisor function up to x. Taking the mean has the eﬀect of ‘smoothing out’ the jumps in the divisor function. But just how well-behaved is this average? Do we get some sort of regular behaviour as x becomes very large?

Plots of the sum of the divisor function, D(x) (le), and its mean value, D(x)/x (right).

33

spring 2016

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