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We study extreme values of Dirichlet polynomials with multiplicativecoefficients, namely  \[D_N(t) : = D_{f,\, N}(t)= \frac{1}{\sqrt{N}} \sum_{n\leqslant N} f(n)n^{it}, \]  where $f$ is a completely multiplicative function with $|f(n)|=1$ for all$n\in\mathbb{N}$. We use Soundararajan's resonance method to produce largevalues of $\left|D_N(t)\right|$ uniformly for all such $f$. In particular, weimprove a recent result of Benatar and Nishry, where they establish weakerlower bounds and only for almost all such $f$.

Extreme values of Dirichlet polynomials with multiplicative coefficients