The Brjuno and Wilton functions bear a striking resemblance, despite their very different origins; while the Brjuno function B(x) is a fundamental tool in one-dimensional holomorphic dynamics, the Wilton function W(x) stems from the study of divisor sums and self-correlation functions in analytic number theory. We show that these perspectives are unified by the semi-Brjuno function B_0(x). Namely, B(x) and W(x) can be expressed in terms of the even and odd parts of B_0(x), respectively, up to a bounded defect. Based on numerical observations, we further analyze the arising functions \Delta^+(x) = B^+(x) - 2B_0^+(x) and \Delta^-(x) = W^-(x) - 2B_0^-(x), the first of which is Hölder continuous whereas the second exhibits discontinuities at rationals, behaving similarly to the classical popcorn function.
