Mastering Number Theory: Analytic Number Theory I: Prime‐Counting and the Riemann Zeta Function

Warning: counting primes may induce an irrational obsession-fortunately, unlike π, your sanity here is well within finite bounds.

Prepare to dive headlong into a realm where every prime tells a story, every zero has its moment under the spotlight, and Python code is your trusty sidekick. Penned by an eccentric Math PhD who respects no wall of chalk dust, this textbook-with full Python demos-guides you through:

- Foundations of π(x), θ(x), ψ(x) and their elegant interrelations
- Euler product, M鐽ius inversion & the Mellin transform as you've never seen them
- Analytic continuation, the functional equation & the critical strip's beguiling zeros
- Perron's formula, contour shifting & explicit formulas connecting primes to zeros
- Zero-free regions (de la Vall嶪 Poussin, Vinogradov-Korobov) and their prime-counting payoffs
- Practical Python scripts for the Prime Number Theorem, error-term exploration, prime gaps & short-interval counts
- Conditional frontiers: Riemann Hypothesis, zero-density theorems, pair correlation & more

Why wade through abstract theorems alone when you can execute them? Each chapter marries rigorous mathematics with interactive code that:

• Visualizes dx/log x integrals and li(x) corrections
• Computes ψ(x) and π(x) to astronomical heights
• Demonstrates explicit error bounds O(x e (-c√log x))
• Explores L-functions, Dirichlet series & primes in arithmetic progressions