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Let $a,b,c$ be pairwise coprime integers. Let $\mathcal{P}_1$, $\mathcal{P}_2$ be two sets of prime numbers with positive relative density. We can assume that the primes in each $\mathcal{P}_i$ do not …

Let $\{a_n\}, \{b_m\}$ be two sequences of complex numbers satisfying $|a_n|, |b_m| \leq 1$ for all $m,n \geq 1$. For any positive numbers $N,M \geq 1$ and $\varepsilon > 0$, it is known that the esti …

One of the most important theorems proved in the 19th century is the prime number theorem. Put $\pi(x)$ for the number of prime numbers $p$ satisfying $1 \leq p \leq x$. Then the prime number theorem …

Let $m_1, m_2$ be positive integers (not necessarily co-prime) and let $S_1, S_2$ be a set of congruence classes modulo $m_1, m_2$ respectively. Let $P_1, P_2$ be the set of prime numbers belonging to …

For a positive integer $n$, let $\omega(n)$ be the number of distinct prime divisors of $n$. For non-zero integers $m,n$ let $\left(\frac{m}{n}\right)$ be the Jacobi symbol. For positive $X$, interp …

Let $F(x,y)$ be an irreducible binary form with integer coefficients and degree $d \geq 3$. Denote by $A_F$ the area of the region $$\displaystyle \{(x,y) \in \mathbb{R}^2 : |F(x,y)| \leq 1\}.$$ It …

Let $f(x,y) = ax^2 + 2bxy + cy^2 \in \mathbb{Z}[x,y]$ be a positive definite binary quadratic form with even discriminant. We say that $f$ is primitive if $\gcd(a,2b,c) = 1$ and it is reduced if $0 < …

Let $n \in \mathbb{N}$. Define the radical $R(n)$ of $n$ by $$\displaystyle R(n) = \prod_{p | n} p.$$ In other words, $R(n)$ is the largest square-free number which divides $n$. For an integer $k …

The (in)famous Gauss circle problem asserts that the usual error term for counting integral points inside a circle of radius $R$ centred at the origin can be made much smaller. In particular, by using …

Let $f(x) \in \mathbb{Z}[x]$ be an irreducible quadratic polynomial, and let $0 \leq \alpha \leq \beta \leq 1$ be real numbers. Put $$\displaystyle S_f(\alpha, \beta; d) = |\{v \in \{1, \cdots, d\} : …

By Dirchlet's hyperbola method, one can prove that the average number of divisors of integers $1 \leq n \leq X$ is $\log X$. This question concerns the number of integers $n \leq X$ such that the numb …

In 1933, Kurt Mahler proved the following: Let $F$ be a binary form with integer coefficients which is irreducible over $\mathbb{Q}$ and has degree $d \geq 3$. Then the area $A_F$ of the region $$\di …

Consider the two subsets of the natural numbers $A$, $B$ given by $$A = \{n \in \mathbb{N} \mathrel: \text{$\exists x,y \in \mathbb{Z}$ s.t. $n = x^2 + y^2$}\}$$ and $$B = \{n \in \mathbb{N} : \exists …

The question asks what is known about integer solutions $(\mathbf{x}, \mathbf{y}) = ((x_1, x_2, x_3), (y_1, y_2, y_3))$ to the equation $$\displaystyle x_1^{r-1} y_1^r + x_2^{r-1} y_2^r = x_3^{r-1} y_ …

I read in the following paper of Toth (https://arxiv.org/pdf/1608.00795.pdf) that Ramanujan had obtained the following formula, proved by Wilson in 1922: $$\displaystyle \sum_{n \leq X} \frac{1}{d(n) …