Worked solution: Tao's proof of the Erdős discrepancy problem via logarithmically averaged correlations (2015)

Following the Polymath5 project (reproduced in Section 2 of Tao's paper), a Fourier-analytic and compactness argument converts the vector-valued Erdős discrepancy statement into an equivalent claim about stochastic completely multiplicative functions: a stochastic completely multiplicative function $\mathbf{g}:\mathbb{N}\to S^1$ is a random variable, defined by randomising its prime values (with no independence assumption in the general stochastic formulation), such that $\mathbf{g}(nm)=\mathbf{g}(n)\mathbf{g}(m)$ almost surely. Theorem 1.8 of Tao's paper states that the original conjecture holds if and only if $\sup_n \mathbb{E}\left|\sum_{j=1}^n \mathbf{g}(j)\right|^2 = \infty$ for every such $\mathbf{g}$.

The key technical steps (worked out explicitly with Fourier coefficients and a compactness/Prokhorov argument in Section 2) build, for any hypothetical counterexample $f$ with bounded discrepancy $\le C$, an explicit stochastic completely multiplicative function $\mathbf{g}_X$ approximating $f$ on the range $n\le X$ using the Chinese Remainder Theorem and Fourier duality on $(\mathbb{Z}/M\mathbb{Z})^r$, then takes a weak limit as $X\to\infty$; this reduces "disprove the theorem for some real sequence" to the cleaner, purely probabilistic question "disprove the theorem for some random completely multiplicative function".

This reduction is what lets the rest of the proof bring in powerful tools from multiplicative number theory — designed to study exactly these random or deterministic completely multiplicative functions — that would not directly apply to an arbitrary $\{-1,+1\}$-valued sequence.