Extreme values of Dirichlet polynomials with multiplicative coefficients

2023/03/12 by Max Wenqiang Xu, Xu, Max Wenqiang, Daodao Yang +1 · 5 citations
Mathematics · #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Meromorphic and Entire Functions #Number Theory (math.NT) #Probability (math.PR)

paper · pdf · doi:10.48550/arxiv.2303.06739

openalex publication_date 2023/03/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

Abstract

We study extreme values of Dirichlet polynomials with multiplicative coefficients, namely D_N(t) : = D_{f, N}(t)= (1)/(√(N)) ∑_{n\leqslant N} f(n) n^it, 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 |D_N(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.

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• Large values of derivatives of the Riemann zeta function on vertical homogeneous progressions
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• Large values of Dirichlet polynomials with multiplicative coefficients
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• Large values of Dirichlet polynomials with multiplicative coefficients
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• Large values of character sums with multiplicative coefficients
• Quadratic character sums with multiplicative coefficients
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