Extreme values of Dirichlet polynomials with multiplicative coefficients

We study extreme values of Dirichlet polynomials with multiplicative coefficients, namely $$D_N(t):=D_{f,N}(t)=\frac{1}{\sqrt{N}}\sum _{n⩽N}f(n)n^{it},$$ where $f$ is a completely multiplicative function with $\mathrm{\mid }f(n)\mathrm{\mid }=1$ for all $n\in \mathbb{N}$. We use Soundararajan’s resonance method to produce large values of $\mid D_N(t)\mid$ uniformly for all such $f$. In particular, we improve a recent result of Benatar and Nishry, where they establish weaker lower bounds and only for almost all such $f$.