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

The Elliott conjecture, in its classical form, predicts that for two bounded multiplicative functions $g_1,g_2:\mathbb{N}\to\{z\in\mathbb{C}:|z|\le1\}$, the correlation $\frac1x\sum_{n\le x}g_1(n)g_2(n+1)$ tends to $0$ as $x\to\infty$, unless one of $g_1,g_2$ is "pretentious": close, in a precise averaged sense, to a twisted Dirichlet character $n\mapsto \chi(n)n^{it}$. The unrestricted (non-averaged) version of this conjecture was known to be extremely difficult, blocked by the so-called parity problem that stymies most sieve-theoretic approaches to related questions like the twin prime conjecture.

Tao's key input, Theorem 1.10 in the discrepancy paper (proved in the separate companion paper "The logarithmically averaged Chowla and Elliott conjectures for two-point correlations", arXiv:1509.05422, published in Forum of Mathematics, Pi, 2016), establishes a logarithmically averaged version of exactly this statement: $\frac{1}{\log x}\sum_{n\le x}\frac{g_1(n)g_2(n+1)}{n} \to 0$ as $x\to\infty$, whenever $g_1$ is non-pretentious in a suitable quantitative sense. Averaging with the extra weight $1/n$ and normalising by $\log x$ (rather than $x$) turns out to sidestep the parity problem, at the cost of only proving the weaker logarithmically averaged statement.

Applying this theorem to the bound from the previous step (with $g_1=g_2=\mathbf{g}$, correlating $\mathbf{g}(n)$ with $\mathbf{g}(n+1)$) shows that the assumed boundedness of $\mathbf{g}$'s partial sums is only possible if $\mathbf{g}$ itself is pretentious — i.e. correlates strongly with some $n\mapsto\chi(n)n^{it}$. This is the pivot of the whole argument: an analytic hypothesis about second moments has been converted into a rigid algebraic/arithmetic statement about $\mathbf{g}$'s structure.