Exponential sums over primes with multiplicative coefficients

Abstract

We consider exponential sums of the form

$$\begin{aligned} \sum _{X< p \le 2X}f(p)(\log p) e(p\alpha ), \end{aligned}$$

where the sum runs over the prime numbers $p\in (X, 2X]$ and f is a multiplicative function satisfying certain growth conditions. As a consequence of our result, we consider the coefficients $(a_\pi (n))$ of the standard L-function $L(s,\pi )$ of a cuspidal representation $\pi$ of GL(d) that satisfies the Ramanujan–Petersson conjecture as well as an estimate of the form $\max _{\alpha \in \mathbb {R}}|\sum _{\begin{array}{c} n\le X \end{array}} a_\pi (n) e(n\alpha )|\le X^\eta$ for some $\eta <1$. For such a form, we get that

$$\begin{aligned} \sum _{X<p\le 2X} a_\pi (p)(\log p) e(p\alpha )\ll \frac{\sqrt{q}}{\varphi (q)}X, \end{aligned}$$

where $\alpha$ is a real number such that $\left| \alpha -\frac{a}{q}\right| \ll X^{-1+\frac{1-\eta }{120}}$ for some $q\le X^{(1-\eta )/15}$. Under stronger restrictions and the same conditions on $\alpha$ and a/q, we also prove that

$$\begin{aligned} \sum _{X<\ell \le 2X} a_\pi (\ell )\mu (\ell ) e(p\alpha )\ll X/\sqrt{q}. \end{aligned}$$