The Brjuno and Wilton functions

Abstract The Brjuno and Wilton functions bear a striking resemblance, despite their very different origins; while the Brjuno function is a fundamental tool in one‐dimensional holomorphic dynamics, the Wilton function 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 . Namely, and can be expressed in terms of the even and odd parts of , respectively, up to a bounded defect. Based on numerical observations, we further analyze the arising functions and , the first of which is Hölder continuous whereas the second exhibits discontinuities at rationals, behaving similarly to the classical popcorn function.