The Brjuno and Wilton functions

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 H & ouml;lder continuous whereas the second exhibits discontinuities at rationals, behaving similarly to the classical popcorn function.