Worked solution: Tao's proof of the Erdős discrepancy problem via logarithmically averaged correlations (2015)
The chain of reasoning has closed a full loop: assuming some sequence had bounded discrepancy led, step by step, to a specific random multiplicative function that is forced to disguise itself as a Dirichlet character, and that very disguise is what makes its discrepancy grow without bound after all — contradicting the starting assumption. Since the assumption is impossible, every -valued sequence must have infinite discrepancy, exactly as Erdős conjectured in the 1930s.
Tao's paper, submitted in September 2015 and published in Discrete Analysis in 2016, closed a problem that outlasted Erdős himself (who died in 1996) and the large-scale Polymath5 collaboration that tackled it directly. It stands as one of the clearest demonstrations that deep tools from analytic number theory — built originally for entirely different purposes, like the twin prime problem — can resolve seemingly elementary combinatorial questions.
Steps 4 through 8 together constitute a proof by contradiction: assuming (step 4) that some stochastic completely multiplicative function has bounded second moments of its partial sums leads, via the logarithmically averaged Elliott conjecture (steps 5–6), to the conclusion that must pretend to be a twisted Dirichlet character (step 7); but the same pretentious structure, examined via a generalised Borwein–Choi–Coons construction (step 8), forces 's partial sums to actually grow like , without bound — contradicting the assumption. Hence no such bounded exists, so by the equivalence from step 3, the original Erdős discrepancy conjecture holds for every -valued sequence (and indeed, as Tao's Theorem 1.1 states, for every sequence valued in the unit sphere of any real or complex Hilbert space).
The paper was submitted to arXiv on 17 September 2015 and published in the inaugural volume of the journal Discrete Analysis in 2016 (DOI 10.19086/da.609), a journal specifically created for the arXiv-overlay publication of high-quality combinatorics papers. Tao has described the discovery of the entropy decrement argument used to prove the key Theorem 1.10 as arising directly from his attempt to solve this specific discrepancy problem, illustrating how a concrete, easily stated combinatorial challenge can motivate a genuinely new technique in analytic number theory.
The Erdős discrepancy problem thus joins a short list of conjectures — easy to state, resistant to elementary methods, and ultimately settled only by importing machinery from a seemingly distant area of mathematics.