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In their paper "Exponential Sums with Monomials," Fouvry and Iwaniec study exponential sums roughly of the form $$ \sum_{m_1 \sim M_1} \cdots \sum_{m_r \sim M_r} c_1(m_1) \cdots c_r(m_r) e\...

I'm working on a problem right now in which I need an upper bound for an exponential sum of the form $$ \tag{1} \sum_{N < n \leq 2N} \tau_3(n) e(f(n)), $$ where $\tau_3(n) = \sum_{d_1d_2d_3=n} 1$ ...

In some recent work, the following strange-looking exponential sum arose: $$ \sum_k \sum_r \sum_s e\bigg( \frac{r \bar s^{(r)} \bar k^{(r^2+s^2)}}{r^2+s^2} \bigg). $$ Here $e(x) = e^{2\pi i x}$ as ...

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 ...