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Let $\Lambda \subset\mathbb{R}^n$ be a lattice satisfying $\|x-y\|_2^2 \in \mathbb{Z}$ for all $x,y\in\Lambda$. How small can $\text{vol}(\Lambda)=\det(\Lambda)$ be? For example, in dimension $2$, the …

I was reading about some of the zero density Theorems for my Analytic Number Theory Topics course. While looking over some more complicated results and proofs a few simple questions came up: First, …

Hello, I am wondering if there is a simple asymptotic formula for $$\sum_{p\leq x}\frac{\left(\log p\right)^{k}}{p},$$ where $k\geq0$ is some integer. If $k$ is $0,$ by using the Prime Number Theor …

The best unconditional result bounding prime gaps is due to Baker, Harman and Pintz, and states that for any sufficiently large $n$, the interval $$[n,n+Cn^{0.525}]$$ contains a prime, for some consta …

Let $\psi(x)=\sum_{n\leq x} \Lambda(n)$ be the weighted prime counting function. I am trying to evaluate the integral $$\kappa:=\int_{1}^{\infty}\frac{\psi(x)-x}{x^{2}}dx$$ in several different ways. …

I am trying to find a formula for the following integral for non-negative integer $k$: $$\int_1^{\infty}\frac{\{u\}}{u^{2}}\left(\log u\right)^{k}du.$$ My first thought was to use the formula $$\ze …

Let $\mathcal{H}=\left\{ h_{i}\right\} _{i=1}^{k}$ be an admissible set, and define $$\pi_{\mathcal{H}}(x)=\left|\left\{ n\leq x\ :\ \exists\ i,j\leq k,\ i\neq j\ \text{such that both }n+h_{i},\ n+h_ …

For $q,a$ relatively prime, let $\pi(x,q,a)$ denote the number of primes less than $x$ which are congruent to $a$ modulo $q$. The Brun-Titchmarsh theorem states that $$\pi(x,q,a)\leq \frac{(2+o(1))x}{ …

Let $\pi(x)=\sum_{p\leq x}$ denote the prime counting function. A well known result of Baker, Harman, and Pintz on prime gaps states that for $x\geq y\geq x^{0.525}$ we have that $$\pi(x+y)-\pi(x)\gg …

The Bombieri-Vinogradov Theorem states that given $A>0$, there exists $B>0$ such that for $Q=\sqrt{x}\left(\log x\right)^{-B},$ we have $$\sum_{q\leq Q}\max_{y\leq x}\max_{\begin{array}{c} a\text{ mo …

Let $\chi$ be a Dirichlet character, and define $\phi_\chi (n)$ so that it satisfies $$\sum_{n=1}^\infty \phi_\chi (n)n^{-s}=\frac{\zeta(s-1)}{L(s,\chi)}.$$ In other words $$\phi_{\chi}(n)=\sum_{d| …

The Borel-Cantelli lemma is very frequently used to give a heuristic for whether or not certain statements in number theory are true. For example, it gives some evidence that there are finitely many …