Finiteness of the critical 4/3-variation of Riemann’s function

Establish that Riemann’s function R(x)=\sum_{n=1}^{\infty}\sin(\pi n^{2}x)/n^{2} has finite critical 4/3-variation.

Background

The paper proves that Riemann’s function has finite p-variation for every p>4/3 and infinite p-variation for every p<4/3, leaving only the critical exponent p=4/3 unresolved. The authors introduce the complex-valued function $$\Phi(x)=\sum_{n=1}^{\infty}e^{\pi i n^{2}x}/(\pi i n^{2})$$, whose p-variation is finite exactly when that of R is finite for p>1. They obtain a logarithmic upper bound for 4/3-variation sums over arbitrary disjoint interval families, but this does not establish uniform boundedness for all such families; proving finiteness at the critical exponent would resolve the conjecture.