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

Combining the shifted second-moment bound from step 4 with Theorem 1.10 (step 5–6), Tao shows in Section 4 of his paper ("A generalised Borwein–Choi–Coons analysis") that, with probability $1-O(\varepsilon)$, there exists a Dirichlet character $\boldsymbol{\chi}$ of period $\mathbf{q}=O_\varepsilon(1)$ and a real number $\mathbf{t}=O_{C,\varepsilon}(X)$ such that $\sum_{p\le X}\frac{1-\operatorname{Re}\,\mathbf{g}(p)\overline{\boldsymbol{\chi}(p)}p^{-i\mathbf{t}}}{p}\ll_\varepsilon 1$; this says precisely that $\mathbf{g}$'s values at primes agree, in an averaged sense, with those of $\boldsymbol{\chi}(n)n^{i\mathbf{t}}$. A careful further argument (Lemma 4.1, using the Vinogradov–Korobov zero-free region for Dirichlet $L$-functions to rule out two very different candidate values of $\mathbf{t}$ both fitting well) additionally pins down $\mathbf{t}$ to be small, $\mathbf{t}=O_{C,\varepsilon}(X)$, and unique up to this error.

Writing $\mathbf{g}(n) = \tilde{\boldsymbol{\chi}}(n) n^{i\mathbf{t}} \mathbf{h}(n)$ for an explicit completely multiplicative $\tilde{\boldsymbol{\chi}}$ agreeing with $\boldsymbol{\chi}$ off the primes dividing $\mathbf{q}$, the bound above translates into $\left|\sum_{p\le X}\frac{1-\operatorname{Re}\,\mathbf{h}(p)}{p}\right|\ll_\varepsilon 1$: the "error" factor $\mathbf{h}$ is itself forced to pretend to be the constant function $1$.

What remains is to show that a genuine $\{-1,+1\}$-valued (or unit-circle-valued) completely multiplicative function that pretends this strongly to a twisted Dirichlet character cannot, after all, have bounded second moment — exactly reversing the near-counterexample from the second step.