I am looking for the best upper bound on $$\sum_{\substack{d | n\ d \geq N}} 1.$$ I know that $$ d(n) = \sum_{\substack{d | n}} 1 \leq e^{O(\frac{\log n}{\log \log n})}. $$ For …

This is my third question on this site regarding Montgomery's conjecture -- and I apologize if this is too much -- but I am still not understanding well why this conjecture is beli …

Inspired by a blogpost by Scott Morrison and ongoing discussion there I decided to create this community wiki to track progress on the original bound of Yitan Zhang. The original …

One of Fermat's theorems states that if $p = 4n + 1$ for some integer $n$, then $p$ can be expressed uniquely as a sum of two squares, $p = a^2 + b^2$. I am working on a problem a …

Here is a link on the internet: https://www.dropbox.com/s/su3uak2a057yrqv/YitangZhang.pdf Can someone teach me how to use trivial estimation to reach (6.1) on page 24? Namely, how …

Iwaniec and Kowalski, in their famous book Analytic Number Theory states a strong form of the effective Chebotarev density theorem page 143, and prove it assuming both GRH for Arti …

What are some techniques and theorems of analytic number theory that have proved useful outside of number theory?

On the bottom of the page 399. of Iwaniec and Kowalski's Analytic Number Theory, the authors claim that $$h(t)=\int_{\mathbb H}k(i,z)y^s d\mu (z)$$ yields $$h(t)=\sqrt{\pi}\frac{\ …

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 …

There is an elementary statement that I believe I have read somewhere, but I can't remember where. I'd like to know if the statement is correct (in which case it is surely standard …

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" …

Construct a family of sets $A_n$ such that $$|A_n|=\Theta\left((\log n)^2\right)$$ and the elements of $A_n$ are chosen uniformly at random mod $n$. Say that a set $S$ represents …

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 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 …

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 …