Let $\alpha$ be any positive irrational and $\beta$ be any positive real. We have the following results. H. Weyl (1909): The fractional part of the sequence $\alpha n$ is equidist …

I am interested in the zero-set of a homogeneous function $f(x_1, \cdots, x_n)$, where $f$ is not necessarily a polynomial. In particular, I would like to know if there are any gen …

What is the current best result on the greatest lower bound on gaps between P2 almost primes where P2 represents a prime or the product of two semi-primes?

I have an equation that I think should not have too many solutions, but I don't see a way to argue this. Given $a, b, c, N \in \mathbb{N}$, how many positive integer solutions $x, …

Using a simple java application, I have noticed that for $x > 25$: $$\psi\left(\frac{x}{5}\right) \ge \psi\left(\frac{x}{3}\right) - \psi\left(\frac{x}{4}\right)$$ where: $$\psi …

Let $a$ be an integer, $p$ a prime (much) greater than $a$, and $\chi$ a Dirichlet character. There is an abundant literature on the sums $$S(\chi,a)=\sum_{i=1}^a \chi(i),$$ called …

There is an abundant literature, and even here on MO no shortage of questions, on the question of the smallest prime primitivee root modulo $q$ (where $q$ is a prime, or more gener …

Let $d(n)$ denote the number of divisors of $n$. Is it known that the series $$\sum_{p \text{ prime}} \frac{1}{d(p-1)}$$ diverges? This would follow immediately from the Sophie Ge …

Does anyone know of the best estimates for $n_i$ and $n_{i+1}-n_i$ where $n_i$ is the $i-$th $y-$smooth number? The best I could find was Tijdemann's estimate for the gap in terms …

The short version of my question goes: What is known to follow from the existence of Siegel zeros? A longer version to give an idea of what I have in mind: The "expectional zeros" …

In regard to the characteristics of certain "explicit formulae" arising in number theory, I am pondering the connection between the rate of convergence of series and the asymptotic …

Let $d(n)$ denote the largest divisor of $n$ less than $\sqrt{n}$. Are there good lower bounds for $d$ that hold for almost all natural numbers? More precisely, is there a functi …

I was reading through Jitsuro Nagura's proof that there is always a prime between $x$ and $\frac{6x}{5}$ when $x \ge 25$. In the paper, he uses the following bounds for the second …

The resolution of the Diophantine equation $$m! = n(n+1)$$ was asked on M.SE. My intuition says that this cannot be solved by elementary means - apologies if I am mistaken. I felt …

The Jacobi sum of $n$ multiplicative character $\chi_1,\dots,\chi_n$ on a finite field $\mathbb F_q$ is defined as $$J(\chi_1,\dots,\chi_n) = \sum_{x_1,\dots,x_n \in \mathbb F_q, …