You can count on Dirichlet James Cann
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.