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Let $\alpha \in I$ where $I$ is some closed interval that does not contain $0$. I am interested in upper bound for $$ M(\alpha) = \#\{ 1 \leq n \leq N: \| \alpha n^2/N \| < 1/N \} $$ where $N$ is a …

In Montgomery's "Ten Lectures on the Interface Between Number Theory and Harmonic Analysis" a bound for the Fourier coefficients of the Selberg polynomial $S^+_K$ is obtained by using what he calls Va …

Vaaler's polynomial is defined $$ V_K(x) = \frac{1}{K+1}\sum_{k=1}^K\left(\frac{k}{K+1} - \frac12\right) \Delta_{K+1}\left(x - \frac{k}{K+1}\right) + \frac{1}{2 \pi (K+1)}\sin 2 \pi (K+1) x - \frac{1} …

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 $$ …

Let $f, g \in \mathbb{Z}[x]$ be coprime polynomials. I am interested in an upper bound for $$ N(B) = \# \{ x \in [-B, B] \cap \mathbb{Z}: f(x)\mid g(x) \}. $$ I assume there must be something known …

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 $\alpha > 0$ be a real irrational algebraic number and $c > 0$. I am interested in the following question. Does there exist a sequence $(x_i,y_i) \in \mathbb{Z}^2$ such that $$ \lim_{i \rightarrow …

Let $n_1, n_2, ...$ be a sequence of natural numbers such that $\{n_i: i \in \mathbb{N}\}$ as a set is all of natural numbers. Let $k$ be a positive integer. Is is possible to obtain a lower bound of …

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 …

Let $0 \neq \alpha \in [0,1]$ and $q$ a positive integer. Let $||.||$ denote the distance to the closest integer and define $$ N_i(q) = \sum_{ \substack{ -q/2 \leq n \leq q/2 \\ \frac{i}{q} \leq || \a …

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 …

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 …

A weaker version of the Riemann hypothesis is the claim that if $\zeta(s) = 0$ then $Re(s) \leq 1 - h$ for some constant $h> 0$. What would the consequences be of a result of this type?

I am confused with what seems to be a standard notation in analytic number theory and I'd appreciate any clarification. I am interested in the zero density estimates, for example link.springer.com/ar …

Let $T$ be a triangle in $\mathbb{R}^2$ defined by $y = \alpha x$, $y = \beta$ and $x = \gamma$ where $\alpha, \beta, \gamma \in \mathbb{R}_{>0}$. I am interested in obtaining an estimate for the numb …