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You can count on Dirichlet James Cann

I

the days before email, mathematicians relied upon pen, paper and the postman to share ideas and communicate fiendish numerical taunts. An excited Dirichlet wrote to Kronecker in 1858:

“... that sum, which I could only describe up to an error of order √x at the time of my last letter, I’ve now managed to home in on signiﬁcantly.”∗

Dirichlet’s sum is associated to the divisor function d(n): the number of distinct divisors of the natural number n. The number six is divisible by 1, 2, 3 and 6, so d(6) = 4. Four has divisors 1, 2 and 4, and so d(4) = 3. It might seem odd that a great mathematician was troubled by a quantity whose description amounts to a good grasp of ‘times tables’—still odder that he was confessing to a distinguished contemporary that the damned thing had caused him grief. It is not hard (although perhaps time-consuming) A plot of the divisor function, d(n). to work out the value of d(n) for any given natural number n. But what if we want to provide a formula to describe it? Where to begin? ∗

Translated from the original using the author’s indelicate German.

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