Extreme values of Dirichlet polynomials with multiplicative coefficients

Abstract

We study extreme values of Dirichlet polynomials with multiplicative coefficients, namelyDN(t):=Df,N(t)=1N∑n⩽Nf(n)nit, where f is a completely multiplicative function with |f(n)|=1 for all n∈N. We use Soundararajan's resonance method to produce large values of |DN(t)| 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.