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Let $F_1(\mathbf{x}, \mathbf{y}), \ldots, F_r(\mathbf{x}, \mathbf{y})$ be bihomogeneous polynomials with rational coefficients with bidegree $(d_1, d_2)$, which means $$ F_i( s x_1, \ldots, s x_{n_1} …

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 $A_1,..., A_s \in M_n(\mathbb{R})$ be symmetric matrices and suppose they are linearly independent over $\mathbb{R}$. This means that $$ m = \min_{(c_1, ..., c_s) \in \mathbb{R}^s \backslash \{0\} …

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 …

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 …

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 …

I am having trouble proving the following statement, which I think is true (and possibly very basic). Let $M$ be a real differentiable manifold of dimension $(n-1)$ sitting inside $\mathbb{R}^n$. Let …

Let $\alpha \in \mathbb{R}$ and $N$ a positive integer. I am interested in the quantity $$ D(\alpha, N) := \# \{ n \in [1, N]: \| n \alpha \| < 1/N \}, $$ $\| x \|$ denotes the distance to the closes …

Let $f(x): \mathbb{R} \to \mathbb{R}_{\geq 0}$ be a smooth function. I am interested in estimating sums of the form $$ \sum_{ A < n \leq B } \{ f(n)\} $$ where $\{ c \}$ denotes the fractional part of …

Let $\mu(n)$ be the Mobius function. Let us define $\mu^+(n)$ to be $\mu(n)$ if $\mu(n)>0$ and $0$ otherwise. Is there a known asymptotic formula for $$ \sum_{n \leq N} \mu^+(n), $$ and similarly f …

Vaughn's identity is a useful way to decompose the von Mangoldt function $\Lambda(n)$ into Type I and Type II components, and this is used in many problems involving prime numbers. I was wondering if …

Let $r(B)$ be the number of integers $1 \leq n \leq B$ such that $n = x^2 + y^2$ for some $x, y \in \mathbb{Z}.$ Then it is a known theorem of Landau that $$ r(B) \sim C \frac{B}{\sqrt{\log B}} $$ f …

Let $\Lambda$ be the von Mangoldt function. I think the following is known to hold under GRH: Given any $q \geq 1$, $(a,q)=1$, and $X \geq 1$, we have $$ \sum_{\substack{1 \leq n \leq X \\ n \equiv …

Let $$\psi(x;q,a)=\sum_{n\leq x\atop n\equiv a\pmod q}\Lambda(n)$$ where $\Lambda$ denotes the von Mangoldt function and $\phi$ to be Euler's totient function. Then the Siegel-Wafisz theorem states t …

A set $S \subseteq \mathbb{Z}/p\mathbb{Z}$ is called a Sidon set if given $a, b, c, d \in S$ and $a+ b = c+ d$, then $\{a, b\} = \{c,d\}$. I was interested in knowing about the largest possible Sidon …