# Analytic Proof of the Prime Number Theorem

Taking down $\psi(x)$ by complex integration

# The Prime Number Theorem and Its Equivalences

Demystifying the counting function $\pi(x)$

# Borel-Caratheodory Lemma and Its Application

Tactical estimation of logarithmic derivatives

# Riemann-von Mangoldt formula for $\zeta(s)$

A powerful tool for us to study nontrivial zeros of $\zeta(s)$

# Alternative Approach to Redefine $\zeta(s)$

Functional equation but the hardcore approach

# Bernoulli Numbers and Their Associated Polynomials

One of the most brilliant tools in asymptotic analysis!

# Mertens' Formula

A brilliant association between primes and logarithms

# A Complete Investigation on Fourier Transform

A comprehensive derivation of different forms of Fourier transform

# Jensen's Formula and Its Implications

Does maximum modulus tells distribution of roots?

# Stirling's Formula

An efficient way to calculate factorial?

# Analytic Continuation of the Riemann Zeta Function

Finally, we can catch a glimpse of the million-dollar math puzzle.

# Factorial, Gamma Function, and More

One of the most commonly used way to implement factorials for complex numbers

# Riemann-Stieltjes Integration and Asymptotics

Summation, but calculus approach

# Möbius Inversion and Beyond

Arithmetic functions are fun!

# Use Pycharm to Emulate Vim

What will you do if you don't know what vim is?