Joni's Math Notes

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...

In this post, we derive a simple form of Brun's combinatorial sieve that is already superior to the Eratosthenes-Legendre sieve and ...

In this post, we present the basic idea of sieve methods and derive the simple Eratosthenes-Legendre sieve. Subsequently, there will be post...

In this post, we prove several results about prime sums that are not much weaker than what follows from the prime number theorem, but the pr...

In this post, we give an application of Fourier analysis to combinatorics, more precisely to Ramsey theory. In Ramsey theory , a typical res...

In this post, we give a proof of the law of quadratic reciprocity based on discrete Fourier analysis and more precisely Gauss sums. This cel...

In this post, we consider trigonometric series that do not necessarily arise from Fourier series. In particular, we consider lacunary trigon...

In many contexts, it is important to know whether a polynomial is irreducible. In algebraic number theory, the properties of an algebraic n...

In this post, we discuss divergence results of Fourier series; this previous post was about convergence results. We showed earlier that qui...