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Let $f$ be a $1$-bounded multiplicative function, and suppose that $|\alpha-a/q|\leq 1/(qx)$, where $(a,q)=1$ and $q\leq x$. We prove that $$ \left|\sum_{n\leq x}\frac{f(n)}{n}e( n\alpha)\right| \leq \left(\frac{4}{\sqrt{6}}+o(1)\right)\frac{\log x}{\sqrt{q}}+O(\log q). $$ Here the constant $4/\sqrt{6}$ is optimal. Thi...
Let $f:\mathbb{N}\to{z\in\mathbb{C}:|z|\leq1}$ be multiplicative, and suppose that $\alpha$ has a rational approximation $a/q$ with $|\alpha-a/q|\leq 1/(qx)$ and $q\leq x/((\log x)^2(\log\log x)^3)$. We prove that $\left|\sum_{n\leq x}f(n)e(n\alpha)\right|\leq \left(\frac4{\sqrt6}+o(1)\right)\frac{x}{\sqrt q} +O\left(\...
Let $f$ be an extended Rademacher random multiplicative function (RMF). We show that, for every fixed deterministic function $V(x)$ tending to infinity, almost surely there are arbitrarily large $x$ for which \[ \sum_{n\leq x}f(n) \geq \frac{\sqrt{x}(\log\log x)^{1/4}}{V(x)}. \] The corresponding negative fluctuation h...
We prove, for every fixed integer $k\ge 4$, the correct order of magnitude for the $k$th moments of the function that counts the number of representations of an integer as sums of two prime squares. The upper bound for $k=4$ was previously known up to $\log\log\log x$, and the lower bound for $k\ge 4$ was only known co...
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