chalkdust notes has been found to back up this claim. Perhaps it was a simply a taunt aimed at Kronecker? Perhaps he made a mistake. In any case it gave birth to the Dirichlet divisor problem:

Determine the smallest value α such that Δ(x) = O(xα+ε ) as x → ∞ which holds true for all ε > 0. The “ε condition” is a neat way of saying that we are only interested in determining the smallest possible α up to a fixed power of log x. The logarithm grows more slowly than any positive power of x. So for instance x 2 (log x)7 = O(x 2+ε ) for every ε > 0.

The other end up The first fiy years aer Dirichlet’s leer saw lile improvement. Progress was slow and the going was tough. Together with Lilewood, Hardy decided to take a diﬀerent tack to the ‘O’ approach. Noting the diﬀiculty of containing the error in a big-O envelope, their idea was to “aack the problem from the other end”, introducing the notion of ‘Ω’ results. They sought to display positive functions g(x), for which there exists a constant K such that the multiple Kg(x) is exceeded by |Δ(x)| for arbitrarily large values of x. By finding such a g of as large an order of magnitude as they could, they were proving that the divisor sum error Δ(x), was at best O(g(x)). In this way, they could install the lower bound of an interval for possible α. The year 1915 saw Hardy prove an ‘Ω’ result showing that α > 41 . When read alongside Dirichlet’s hyperbola result we surmise that α ∈ [ 14 , 12 ].

Hot oscillations Since around 1922, the sharp end of research has involved playing with a daunting representation of the error: ( ) ∞ √ x1/4 ∑ d(n) 1 Δ(x) = √ cos 4π nx − π . (6) 4 π 2 n = 1 n 3/4 Deriving such an expression is no mean feat. As illustration: we can encode the divisor function in a Dirichlet series, which turns out to equal the square of the Riemann-zeta function. As a meromorphic function, the techniques of complex analysis can be put to work. Mellin transforms, asymptotic formulæ for Bessel functions and contour integration all feature in the derivation of the equation for Δ(x) above. Because of the oscillatory nature of the cosine, researchers have used this handy expression to provide both lower bounds and upper bounds for α, turning that infamous phrase: “It’s not the size, it’s what you do with it” on its head. Here, it’s what you do with it that indicates its size: ‘Ω’ or ‘O’. 37

spring 2016

Chalkdust, Issue 03

Popular mathematics magazine from UCL