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 $\left|\sum_{n\leq x}f(n)e(n\alpha)\right|\leq \left(\frac4{\sqrt6}+o(1)\right)\frac{x}{\sqrt q} +O\left(\frac{x}{\log x}\right)$, whereas complete multiplicativity improves $4/\sqrt6$ to $1$. Both constants are optimal. In particular, this confirms a folklore conjecture on exponential sums with multiplicative coefficients motivated by the work of Montgomery and Vaughan (1977). We also obtain logarithmically weighted estimates and give counterexamples to related conjectures.
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 $f$ 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...
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...
Nicolas Robles, Alexandru Zaharescu, Dirk Zeindler· 0 citations
Let $n\ge2$ be a fixed integer, and let $K,x,y$ be positive integers satisfying $2\le K\le y<x$. Consider a polynomial \[ f(u)=\alpha u^n+\alpha_{n-1}u^{n-1}+\cdots+\alpha_1u+\alpha_0 \] with real coefficients. We study the quantity \[ V_n(K,x,y) = \sum_{k=1}^{K} \left| \sum_{x-y<p\le x}e(kf(p)) \right|, \qquad e(t)=e^...
We prove that every set $\mathcal{B}\subseteq\mathbb{Z}_{\ge0}$ for which $\mathbb{P}+\mathcal{B}$ contains all sufficiently large integers satisfies $B(X)\gg\log X\log\log X$, where $B(X)=\#(\mathcal{B}\cap[0,X])$ and the implied constant is absolute. This proves a quantitative form of Ruzsa's Conjecture 2 and gives a...
Let $\mathcal{H}_k$ denote the set of normalized holomorphic Hecke cusp forms of weight $k$ for the full modular group $ \mathrm{SL}_2(\mathbb Z)$. For each $f\in\mathcal{H}_k$, let $\lambda_f(n)$ denote the corresponding eigenvalue of the normalized Hecke operator $T_n$. We prove nontrivial upper bounds for \[\sum_{f...
Max Wenqiang Xu, Jun-Ren Zheng· 0 citations
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