Curiosities of the D Function: Playing with a New Toy

Abstract

A new mathematical tool is best known by playing with it. The D function, $D_A(s)=\sum_n(a_{n+1}-a_n)\,a_n^{-s}$, was introduced to read the staircase of the primes. Here we apply it to things far beyond its purpose, and listen to what it says. A sequence that swings ever more violently, $1,-1,2,-2,\ldots$, turns out to have the Riemann zeta function as its D function, while the alternating squares give an entire function built from Dirichlet's eta. The orbit of $27$ under the Collatz map keeps a ledger of its rises and falls in $D(1)$; Recamán's sequence advances at a steady pace beneath its swings; and Kolakoski's sequence, whose balance of ones and twos is an open problem since 1965, can be restated as a property of its D function, which the data confirm to seven decimal places. The gaps of the primes return towards their typical size more strongly than their own shuffle; the digits of $\rho$ and $\eta$, mirror images of each other, speak with different voices but share the same continued fraction. Turned towards its own family, the D function finds that the zeros of the zeta function repel each other while the primes cluster, and that the pole of the zeta function is inherited by its second generation. A stumble along the way, an equation that was not there, reminds us that almost is not exact. Everything here is meant to be looked at rather than proved.