Background

This section provides mathematical foundations for the experiments.

• Cheat sheet
• Arithmetic functions refresher
• Average orders and the Erdős–Kac viewpoint
• Carmichael numbers
• Carmichael’s \(\lambda(n)\) function refresher
• Dirichlet characters refresher
• Dirichlet convolution refresher
• Dirichlet eta function η(s)
• Dirichlet \(L\)-functions refresher
• Divisibility and modular arithmetic (Phase 2 core)
• Divisor functions \(d(n)\) and \(\sigma_k(n)\) refresher
• Euler’s prime-generating polynomial refresher
• Euler’s totient function \(\varphi(n)\) refresher
• Explicit formulas: primes ↔ zeros
• Exploratory visualizations for arithmetic functions
• Factorization pipelines (trial division + Pollard rho)
• Fermat numbers
• Gauss sums refresher
• Gram points and zero counting
• Hardy’s Z-function and the critical line
• Heegner numbers
• Jordan totient \(J_k(n)\) refresher
• Landau’s problems refresher
• Liouville function \(\lambda(n)\) refresher
• Mersenne numbers and primes refresher
• Möbius function \(\mu(n)\) and Mertens function \(M(x)\) refresher
• Partition function \(p(n)\) refresher
• Perfect numbers refresher
• Pretentious number theory refresher
• Primality testing: guarantees, error bounds, and what to report
• Prime counting approximations: π(x), Li(x), and R(x)
• Prime counting: explicit bounds (not just asymptotics)
• Prime-factor counting: \(\omega(n)\) and \(\Omega(n)\) refresher
• Prime number races refresher
• Prime numbers refresher
• Dirichlet’s theorem and PNT(AP) in the form used by the experiments
• Primorials
• Quadratic polynomials (algebraic) refresher
• Riemann zeta function ζ(s)
• Semiprimes
• Taylor series refresher
• von Mangoldt \(\Lambda(n)\) and Chebyshev functions refresher
• Wieferich primes