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We prove a folklore conjecture on exponential sums with multiplicative coefficients: if $f$ is multiplicative with $|f(n)|\le1$, $(a,q)=1$, $|\alpha-a/q|\le q^{-2}$ and $q\le N/(\log N)^2$, then \[ \sum_{n\le N}f(n)\operatorname{e}(n\alpha)\ll\frac N{\log N}+\frac N{\sqrt q}. \] Complete multiplicativity is not require...
Let $f$ be multiplicative, with $|f(p)|\le A$ at primes and $\sum_{n\le x}|f(n)|^2\le A^2x$ for every $x\ge1$. If $|\alpha-a/q|\le q^{-2}$, $(a,q)=1$, and $3\le R\le q\le N/R$, we prove \[ \sum_{n\le N}f(n)\operatorname{e}(n\alpha) \ll_A \frac{N}{\log N} +\frac{N}{\sqrt R}\sqrt{\log\log(3R)} \] with effective implied c...
We construct finite Heath-Brown-type identities for the fractional powers $\zeta(s)^{\pm a/b}$ of the Riemann zeta-function, for every reduced fraction $a/b$ with $0<a/b<1$, from Newton's binomial series in the algebra of arithmetic functions, and we use them to prove the Vinogradov-quality bound $\sum_{n \le x} d_{\pm...
Let $\mu$ be the M\"obius function and $e(t)=e^{2\pi it}$. We prove that if $N\ge2$, $\alpha\in\mathbb{R}$, $(a,q)=1$, and $|\alpha-a/q|\le q^{-2}$, then \[\bigg|\sum_{n\le N}\mu^2(n)e(\alpha n)\bigg|\ll\left(\frac Nq+q\right)(\log 2N)^5, \] with an absolute implied constant, and we deduce the corresponding estimate on...
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