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arXiv · 2503.08206

The Brjuno and Wilton Functions

Claire Burrin·Seul Bee Lee·Stefano Marmi
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Abstract

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 $Δ^+(x) = B^+(x) - 2B_0^+(x)$ and $Δ^-(x) = W^-(x) - 2B_0^-(x)$, the first of which is Hölder continuous whereas the second exhibits discontinuities at rationals, behaving similarly to the classical popcorn function.

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Claire Burrin, Seul Bee Lee, Stefano Marmi. 2025-03-11. The Brjuno and Wilton Functions. https://arxiv.org/abs/2503.08206

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