Bounds for Exponential Sums With Random Multiplicative Coefficients

Abstract For $f$ a Rademacher or Steinhaus random multiplicative function, we prove that $ \max _{\theta \in [0,1]} \frac{1}{\sqrt{N}} \bigl | \sum _{n \leq N} f(n) \mathrm{e} (n \theta ) \bigr | \gg \sqrt{\log N}$, asymptotically almost surely as $N \rightarrow \infty $. Furthermore, for $f$ a Steinhaus random multiplicative function, and any $\varepsilon> 0$, we prove the partial upper bound result $ \max _{\theta \in [0,1]} \frac{1}{\sqrt{N}} \bigl | \sum _{\substack{n \leq N \\ P(n) \geq N^{0.8}}} f(n) \mathrm{e} (n \theta ) \bigr | \ll{(\log N)}^{7/4 + \varepsilon }$, asymptotically almost surely as $N \rightarrow \infty $, where $P(n)$ denotes the largest prime factor of $n$.