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In https://academic.oup.com/blms/article-abstract/20/2/121/266256?redirectedFrom=fulltext Vaughan shows the following bounds for the $L^1$-mean of the exponential sum over primes $$\sqrt x\ll \int _0^...

Define $e(\theta)=e^{2\pi i\theta}, \theta\in [0,1]$, $P_n=\{p_1,p_2,...,p_n\}$ are the first $n$ primes, $\|f\|_1=\int_{[0,1]}|f(\theta)|d\theta$. Problem 1. is it true for all fixed $m\in \mathbb{N^...

Recall that for the classical Kloosterman sum $$ K(a,b,p^t):= \sum_{x \in (\mathbb{Z}/ p^t \mathbb{Z})^* } \psi \left(\frac{ax+bx^{-1}}{p^t} \right),$$ where $\psi(x)=e^{2\pi ix}$, $a,b,t$ are natural ...

I would like to know whether it is possible to obtain cancellation in the sum $$\sum_{p \leq X} e^{{2\pi iX}/{p}}$$ where $X$ is a real number that goes to $\infty$, and $p$ denotes a prime number.

Let $1 \leq i \leq d$, $q \in \mathbb{N}$, and $0 \leq a_{i} < q$. Let $$ \mathfrak{M}^{(i)}_{a_{ i}, q} (C) =\{ \beta_{i} \in [0,1) : | \beta_{i} - a_{i}/q | \leq (\log X)^{C} X^{-i} \} . $$ We ...

Fix $\alpha \notin \mathbb{Q}$. I would like to know a reference that shows $$\mathbb{E}_{n\leq N} \Lambda(n) e^{2\pi i \alpha n} \to 0,$$ as $N$ tends to infinity. I am familiar with Vinagradov's ...