Joni's Math Notes

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...

In this post, we consider cyclotomic polynomials, which are a special class of polynomials with various connections to number theory, algebr...

In this post, we discuss some non-trivial convergence results for Fourier series. Convergence questions for Fourier series have been histo...

We derive the basic properties of Fourier series in this post and apply them to prove some elegant sum and product identities, namely a  for...

In later posts, we consider properties of Fourier series (for instance, convergence and divergence results), as well as lacunary trigonomet...

In this post we prove the existence of primitive roots modulo prime powers, using elementary number theory and presenting numerous problems ...