Liouville function: λ(n) = (−1)^Ω(n). λ(1)=1, λ(2)=−1, λ(3)=−1, λ(4)=1 (Ω(4)=2), λ(6)=1 (Ω(6)=2). Completely multiplicative: λ(ab)=λ(a)λ(b) always. The Pólya conjecture (L(n)≤0 for n≥2) is FALSE — first counterexample at n=906,150,257.
How to Use
Enter n:
- λ(n): +1 or −1
- Ω(n): Prime factor count
- L(n): Summatory function
Pólya Conjecture
Pólya (1919) conjectured L(n)=Σλ(k)≤0 for all n≥2. Seemed true for small n. Haselgrove (1958) disproved it. First explicit counterexample: n=906,150,257 where L(n)=1. But L(n) is mostly negative.
Riemann Hypothesis
RH is equivalent to: L(n) = O(n^{1/2+ε}) for every ε>0. If λ-sums grow slower than √n, RH is true. This connects a simple ±1 function to the deepest unsolved problem in mathematics.
Step-by-Step Instructions
- 1Enter n.
- 2Factor n.
- 3Count Ω(n).
- 4Compute λ(n).
- 5Check L(n).