Blogs
On $n$ for which every group of order $n$ is cyclic
It is known that every group of prime order is cyclic. In this article, we present an arithmetical criterion on $n$ to determine whether every group of order $n$ is cyclic.
17 Apr 2024 LinkBrun's sieve and its applications
We derive Brun's sieve and show that every large even integer is a sum of two integers, each having at most 9 prime factors.
25 Feb 2024 LinkOn the factorization of polynomials
We derive Gauss's lemma and Eisenstein's criterion. Then we use them to prove that every cyclotomic polynomial is irreducible over $\mathbb Q[x]$.
30 Jan 2024 LinkOn the coefficients of cyclotomic polynomials
We prove that when $n$ only has two distinct odd prime divisors, the expansion of the $n$'th cyclotomic polynomial $\Phi_n(x)$ only contains terms of the form $\pm x^k$.
22 Jan 2024 LinkOn Legendre's differential equation
We prove that the eigenvalues of the Legendre differential operator are exactly $\ell(\ell+1)$ for $\ell\in\mathbb Z$ and derive expressions for the eigenfunctions.
16 Jan 2024 LinkOn the spectrum of Fredholm integral equations
We give an elementary argument showing that the spectrum of Fredholm integral equations is discrete.
23 Dec 2023 LinkOn the growth order of derivatives
We prove theorems that produce asymptotic bounds for derivatives of functions and discuss some applications and generalizations.
16 Nov 2023 LinkCycloids and the tautochrone problem
We study cycloid and show that it is the unique solution to the tautochrone problem.
03 Nov 2023 LinkOn differentiation under integral signs
We derive a generalization of Leibniz integral rule to higher dimensions
19 Oct 2023 LinkRiemann mapping theorem and Dirichlet's principle
We give an account of Riemann's original proof of his famous mapping theorem and discuss its connections to Dirichlet's principle of the calculus of variations
10 Oct 2023 LinkWhen theta functions meet Dirichlet characters
We study theta functions twisted by Dirichlet characters and present their applications to Dirichlet $L$-functions.
08 Sep 2023 LinkIntroduction to the transformations of elliptic functions
We give an elementary derivation of Jacobi's imaginary transformations and Landen's transformations and discuss their consequences in the theory of elliptic functions.
25 Jul 2023 LinkAsymptotic theory of integer partitions
We present two proofs of the Hardy-Ramanujan asymptotic formula of $p(n)$.
16 Jul 2023 LinkApplications of the axiom of choice
We discuss several important applications of the axiom of choice in analysis and algebra.
12 Jun 2023 LinkWell-ordered sets and the axiom of choice
We launch a systematic investigation into well-ordered sets.
27 May 2023 LinkAbsolute values and Ostrowski's theorem
We investigate all possible versions of absolute values over $\mathbb Q$.
11 May 2023 LinkOn the discriminant of polynomials
We explore connections between polynomial discriminants and polynomial coefficients and apply them to study the elliptic discriminant.
09 May 2023 LinkIntroduction to Weierstrass elliptic functions
We develop Weierstrass's theory of elliptic functions
09 Apr 2023 LinkJacobian elliptic functions and sum of squares
We use Jacobian elliptic functions to study the representations of integers as sum of squares
09 Apr 2023 LinkThe foundation of Jacobian theta functions
We derive definitions of theta functions from Jacobian elliptic functions.
07 Apr 2023 LinkInfinite products for Jacobian elliptic functions
We deduce infinite product representations for Jacobian elliptic functions.
05 Apr 2023 LinkA modern introduction to Jacobian elliptic functions
We introduce the classical Jacobian elliptic functions from a modern perspective.
02 Apr 2023 LinkEuler's pentagonal number theorem and Dedekind eta function
We give a direct, elementary proof of the functional equation for $\eta(\tau)$.
19 Mar 2023 LinkThe distribution of prime divisors and Erdös-Kac theorem
We study the statistical behavior of $\omega(n)$, the number of distinct prime divisors of $n$.
05 Nov 2022 LinkExplicit formula for Gauss circle problem
We express the error term of the Gauss circle problem in terms of Bessel functions
26 Sep 2022 LinkNumber of integers that are a sum of two squares
We investigate a question parallel to Gauss circle problem
23 Sep 2022 LinkImprovement on Gauss Circle Problem
We improve the error term in the asymptotic expansion of $S(x)$ from the last article
16 Sep 2022 LinkRepresenting Integers as Sum of Two Squares
Theory from $\mathbb Z[i]$ is now applied to study $\mathbb Z$
30 Aug 2022 LinkEuclid's Algorithm and Unique Factorization of Gaussian Integers
We prove unique factorization theorem for $\mathbb Z[i]$ using Euclid's algorithm
26 Aug 2022 LinkAnalytic Proof of the Prime Number Theorem
Taking down $\psi(x)$ by complex integration
25 Mar 2021 LinkThe Prime Number Theorem and Its Equivalences
Demystifying the counting function $\pi(x)$
24 Mar 2021 LinkBorel-Caratheodory Lemma and Its Application
Tactical estimation of logarithmic derivatives
16 Feb 2021 LinkRiemann-von Mangoldt formula for $\zeta(s)$
A powerful tool for us to study nontrivial zeros of $\zeta(s)$
19 Jan 2021 LinkAlternative Approach to Redefine $\zeta(s)$
Functional equation but the hardcore approach
05 Jan 2021 LinkBernoulli Numbers and Their Associated Polynomials
One of the most brilliant tools in asymptotic analysis!
26 Dec 2020 LinkA Complete Investigation on Fourier Transform
A comprehensive derivation of different forms of Fourier transform
23 Dec 2020 LinkJensen's Formula and Its Implications
Does maximum modulus tells distribution of roots?
14 Dec 2020 LinkAnalytic Continuation of the Riemann Zeta Function
Finally, we can catch a glimpse of the million-dollar math puzzle.
28 Nov 2020 LinkFactorial, Gamma Function, and More
One of the most commonly used way to implement factorials for complex numbers
13 Nov 2020 Link