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I would like to estimate from above the following sum $$ \sum_{1 \leq x_1 < X} .. . \sum_{1 \leq x_n < X} \frac{\prod_{1 \leq i \leq n } \phi(x_i)}{\mathrm{lcm}(x_1, .., x_n)^a}. $$ $\phi$ is the Eu …

Let $X$, $T$, and $x_0$ be positive real numbers. Consider the region in $\mathbb{R}^2$ defined by $$ xy \leq X, \ \ x_0 \leq x \leq x_0 + T, \ \ \frac{X}{x_0 + T} \leq y. $$ Let $A$ be the area …

I would like to bound an exponential sum of the shape $$ S = \sum_{ab \leq N \\ a, b > U} \Lambda(a) \omega(b) e^{2 \pi i F(ab)}, $$ where $\Lambda$ and $\omega$ are some weights (actually $\Lambda$ …

Let $\delta>0$. I am interested in obtaining a bound for the sum $\sum_{1 \leq x_1, ..., x_n \leq N} \operatorname{lcm}(x_1, ..., x_n)^{- \delta}$ where lcm denotes the lowest common multiple of the n …

Let $\Lambda$ be the von Mangoldt function and $\chi$ a primitive character mod $q$, then we have the explicit formula $$ \sum_{n \leq X} \Lambda(n) \chi(n) = \delta_{\chi} X - \sum_{ |Im \ \rho| \leq …

Let $\mu$ be the Mobius function, $\tau_k(x)$ the number of ways to write $x$ as a product of $k$ natural numbers and $\phi$ the Euler totient function. I would like to obtain an upper bound for $$ …

Heath-Brown's identity states: Let $K \geq 1, z \geq 1.$ Then for any $n < 2 z^K$ we have $$ \Lambda(n) = - \sum_{1 \leq k \leq K} (-1)^k {{K}\choose{k}} \sum_{ \substack{ m_1 \cdots m_k n_1 \cdots n_ …

Let $\alpha$ be a real number and $|| \cdot ||$ be the distance to the nearest integer. I want to find a non-trivial upper bound for $$ \sum_{1 \leq n \leq X} \min ( || \alpha n ||^{-1} , X/n), $$ w …

Let $q \in \mathbb{N}$. I am interested in getting an upper bound for the sum $$ \sum_{(a_1, a_2, a_3, q) = 1} \sum_{\mathbf{h} \in (\mathbb{Z}/q\mathbb{Z})^n }e( \frac{a_1}{q}\ell_1(h_1, \ldots, h_n …

Let $N$ be a large integer and $I = [aN, bN]$ for some $0 < a < b < 1$. Denote by $\chi_I(x) = 1$ if $x \in I$, $0$ otherwise. I was wondering if there exists a smooth function $w$ with the property …

Suppose we have $\alpha \in \mathbb{R}$. Then we know that $$\sum_{1 \leq n \leq X} e(n \alpha) \ll \min \{ X, \|\alpha\|^{-1} \}$$ where $\| \cdot \|$ is the distance to the nearest integer. I wa …

Let $d(n)$ be the divisor function defined by $d(n) = \sum_{m|n} 1$. I am in need of estimate of the following type: $$ \sum_{Q \leq n \leq Q + H} d^2(n) \ll H (\log (Q + H))^T $$ where $T$ can be any …

Bombieri-Vinogradov theorem (taken from Wikipedia) states: Let $x$ and $Q$ be any two positive real numbers with $x^{1/2}\log^{-A}x\leq Q\leq x^{1/2}.$ Then $$\sum_{q\leq Q}\max_{y<x}\max_{1\le a\le …

Let $q \in \mathbb{N}$ and $\chi$ a Dirichlet character mod $q$. Let $F(x)$ be a polynomial with integer coefficients. I was wondering if a bound for the following sum was available or not: $$ \sum_{0 …

I have encountered the following exponential sum and I would like to obtain a non-trivial upper bound for it. I am not quite sure where to start, and I would greatly appreciate any suggestions on how …