Von Mangoldt Λ(n): log(p) if n=p^k (prime power), 0 otherwise. Λ(2)=log2, Λ(3)=log3, Λ(4)=log2, Λ(5)=log5, Λ(6)=0. The Chebyshev function ψ(x)=ΣΛ(n)≈x encodes the Prime Number Theorem. Deep connections to Riemann zeta: −ζ'/ζ = Σ Λ(n)/n^s.
How to Use
Enter n:
- Λ(n): log(p) or 0
- Prime power: p^k detection
- ψ(n): Summatory function
Prime Number Theorem
ψ(x) = Σ_{n≤x} Λ(n) ~ x. This is equivalent to π(x) ~ x/ln(x). The von Mangoldt function 'weights' prime powers by log(prime), smoothing the prime counting function. This makes analytic arguments cleaner.
Zeta Connection
−ζ'(s)/ζ(s) = Σ Λ(n)·n^{−s}. This Dirichlet series encodes all zeros of ζ. The explicit formula for ψ(x) involves a sum over ALL zeta zeros. This is why Λ is central to analytic number theory.
Step-by-Step Instructions
- 1Enter n.
- 2Check prime power.
- 3Compute Λ(n).
- 4Sum ψ(n).
- 5Compare to n.