Dedekind psi: ψ(n) = n · Π_{p|n}(1 + 1/p). Compare to Euler: φ(n) = n · Π(1 - 1/p). So ψ uses + where φ uses −. Always ψ(n) ≥ n ≥ φ(n). Product: ψ(n)·φ(n) = n² · Π(1-1/p²) = n·J₂(n)/n.
How to Use
Enter n:
- ψ(n): Dedekind psi value
- Compare: φ(n) vs ψ(n)
- Factors: Prime decomposition
Modular Forms
ψ(n) = [SL₂(Z) : Γ₀(n)], the index of the congruence subgroup. This connects number theory to modular forms and the geometry of the upper half-plane. Fundamental in the theory of modular curves.
Key Identities
- ψ(n) = Σ_{d|n} μ(d)² · (n/d) = Σ_{d²|n} μ(d) · σ(n/d²)
- ψ(n) · φ(n) = n · J₂(n) / n
- ψ is multiplicative
Step-by-Step Instructions
- 1Enter n.
- 2Compute ψ(n).
- 3See factorization.
- 4Compare to φ(n).
- 5Check identities.