Extremal bounds for Dirichlet polynomials with random multiplicative coefficients

Jacques Benatar; Alon Nishry

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arXiv · 2204.03519

Extremal bounds for Dirichlet polynomials with random multiplicative coefficients

Jacques Benatar·Alon Nishry

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Abstract

For $X(n)$ a Steinhaus random multiplicative function, we study the maximal size of the random Dirichlet polynomial $$ D_N(t) = \frac1{\sqrt{N}} \sum_{n \leq N} X(n) n^{it}, $$ with $t$ in various ranges. In particular, for fixed $C>0$ and any small $\varepsilon>0$ we show that, with high probability, $$ \exp( (\log N)^{1/2-\varepsilon} ) \ll \sup_{|t| \leq N^C} |D_N(t)| \ll \exp( (\log N)^{1/2+\varepsilon}). $$

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Jacques Benatar, Alon Nishry. 2022-04-07. Extremal bounds for Dirichlet polynomials with random multiplicative coefficients. https://arxiv.org/abs/2204.03519

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