Schulte’s Omega Binomial Convolution
Abstract
Schulte’s binomial convolution relates the total and distinct prime-factor counts.
C denotes the complex numbers and N the natural numbers including zero. Omega, written Ω, is A001222, the number of prime factors counted with multiplicity; omega, written ω, is A001221, the number of distinct prime factors. Their difference is A046660. Subtraction in the exponent is natural-number subtraction; omega(n) is at most Omega(n), so no truncation changes this difference. The sum ranges over all positive divisors d of n, including one and n. The slash n/d denotes natural-number division, which is exact on this divisor set. Powers have natural exponents, including the convention 0^0 = 1. Only Schulte’s general x, y conjecture for positive n is asserted here. The case x = 1 - y is the Dressler-van de Lune 1973 result; other OEIS assertions are outside the claim.
Theorem 1.1 (The binomial divisor convolution).
Proof. Machine-checked in Lean as D5/S3/Arith/SchulteOmegaBinomialConvolution.result (✓ std3). ∎
Resolves. Problems/oeis-a001222-schulte-omega-binomial-convolution (proved) by D5/S3/Arith/SchulteOmegaBinomialConvolution.result.
Citation. Werner Schulte (2018). OEIS A001222, the Ω/ω binomial convolution conjecture (x+y)^Ω(n) = Σ_{d|n} x^Ω(d) (x+y)^{Ω(n/d)−ω(n/d)} y^{ω(n/d)}. URL: https://oeis.org/A001222.
Commentary.
Both sides define multiplicative arithmetic functions after setting their values at zero to zero. On each prime power the convolution becomes a finite geometric sum. Its polynomial identity holds even when y or x+y is zero. Equality on prime powers then gives the identity at every positive natural index.
References
- Truth anchor:
D5/S3/Arith/SchulteOmegaBinomialConvolution.result