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It is a well-known fact that the generalized von Mangoldt function, defined by $$\displaystyle \Lambda_k(n) = \sum_{d | n} \mu(d) \left(\log \frac{n}{d}\right)^k$$ vanishes whenever $n$ has more tha …

For every odd prime $q \geq 3$, does there exist a prime $p < q$ and integers $x,y$ such that $$\displaystyle x^2 + py^2 = q?$$ One can easily show that all primes $q \not \equiv -1 \pmod{3}$ can be w …

A notorious question with prime numbers is estimating the gaps between consecutive primes. That is, if $(p_n)_{n \geq 1}$ is the canonical enumeration of the primes, then set $g_n = p_{n+1} - p_n$. It …

Let $R(X)$ be the region defined by $$\displaystyle R(X) = \{(a,b,c) \in \mathbb{R}^3 \colon |b| \leq a \leq c, \, a,c \geq 1, \, a(4ac-b^2) \leq X\}.$$ I want to know how to estimate the sum $$\di …

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 …

For a positive integer $n$ put $\omega(n)$ for the number of distinct prime divisors of $n$. It is a well-known theorem of Erdős and Kac that the probability distribution for the quantity $\displayst …

The Chebotarev density theorem is one of the most celebrated and important results in number theory. We state the following version: for a number field $K$, Galois over $\mathbb{Q}$ with Galois group …

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 …

It is well-known that the Euler $\phi$-function is multiplicative: that is, for co-prime positive integers $m,n$ we have $\phi(mn) = \phi(m)\phi(n)$. Thus it is defined by its values on prime powers. …

For a number field $K/\mathbb{Q}$, put $h_K, R_K$ respectively for the class number (of the ring of integers $\mathcal{O}_K$ of $K$) and the regulator of $K$ respectively. Moreover, put $\zeta_K(s)$ f …

Let $d > 1$ be a fundamental discriminant, and let $K_d = \mathbb{Q}(\sqrt{d})$. Denote by $\varepsilon_d$ the fundamental unit of $\mathcal{O}_{K_d}$, namely the smallest algebraic integer $\varepsil …

The Pólya–Vinogradov inequality asserts that a non-principal Dirichlet character $\chi$ with modulus equal to $q$ satisfies $$\displaystyle \left \lvert \sum_{N < n < N+M} \chi(n) \right \rvert = O \l …

Let $a,q$ be co-prime integers and let $P(a,q)$ denote the set of primes congruent to $a$ modulo $q$. Is it known whether one can give an asymptotic formula for the expression $$\displaystyle \sum_{\ …

The real number given by the absolutely convergent series $$\displaystyle A = \sum_{k=1}^\infty \frac{|\mu(k)|}{k \phi(k)}$$ is known as Landau's Totient Constant. It can be explicitly evaluated to be …