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

Tao’s Section 4 does not assert a literal identity for an arbitrary period $q$. After Proposition 1.11 writes $\mathbf{g}(n)=\tilde{\boldsymbol{\chi}}(n)n^{i\mathbf{t}}\mathbf{h}(n)$ with a residual $\mathbf{h}$ close to $1$, the generalized Borwein–Choi–Coons argument averages over intervals and uses character sums to obtain the probabilistic lower bound $\sup_n\mathbb{E}\left|\sum_{j=1}^n\mathbf{g}(j)\right|^2=+\infty$. This is modeled on the explicit $\tilde\chi_3$ construction, but it is not the same pointwise formula for general $q$.