Joni's Math Notes

Consider a ''random'' process where the next state depends on the previous one but in a chaotic manner - say we consider a p...

A central question in additive number theory is determining whether a given infinite set $A$ of positive integers is an asymptotic basis of ...

We consider some classical theorems in arithmetic Ramsey theory, where one wants to find some monochromatic arithmetic structure (an arithme...

In the previous post , we considered Ramsey theory for graphs. Graphs are however not the only combinatorial structures that are guaranteed ...

Ramsey theory is a branch of combinatorics in which theorems typically state that any sufficiently large structure (e.g. a graph, a set of i...

We derive the Rosser-Iwaniec sieve, which gives both upper and lower bounds for the linear sieve problem that are asymptotically optimal. Li...

When considering primes in arithmetic progressions, one is naturally led to study the zeros of Dirichlet $L$-functions. As will be seen, one...

In this post, we will derive Harman's sieve, which enables one to show that a sparse set $A$ contains the expected proportion of primes ...

Uncertainty principles in Fourier analysis are formalizations of the following principle: A function $f$ and its Fourier transform $\hat{f}$...

In this post, we will derive the large sieve, which is one of the most classical sieves. The large sieve is quite different from combinatori...

In ths post, we prove a theorem of Ingham, which states that there is always a prime on the short interval $[x,x+x^{\frac{5}{8}+\varepsilon}...

In this post, we prove several bounds for rather general exponential sums, depending on the growth of the derivative of their phase function...

We prove here an improved prime number theorem of the form \[\begin{eqnarray}\pi(x)=Li(x)+O(x\exp(-c\log^{\frac{4}{7}}x)),\quad Li(x):=\int...

In this post, we consider Weyl sums, which are exponential sums with polynomial phases; a Weyl sum is defined by \[\begin{eqnarray}f(\alph...

We prove in this post an estimate for the prime exponential sums \[\begin{eqnarray}S(N;\alpha):=\sum_{p\leq N}e(\alpha p)\end{eqnarray}\] u...

The ternary Golbach problem is the assertion that every odd integer $N\geq 7$ is the sum of three primes. It was first proved for large enou...

In this post, we consider character sums, which are one of the central objects in the discrete Fourier analysis of the integers modulo $q$. ...

In this post, we derive Selberg's upper bound sieve. Selberg's sieve is a combinatorial sieve based  on the simple but immensely use...

We present some classical results about the Fourier transform in $\mathbb{R}^n$ in this post, and some of them will be applied in later post...

We improve the Brun sieve from the previous post to allow us to consider almost primes of various forms. We will prove for example the resul...