Schulte’s Liouville and Cube-Mobius Identity

Abstract

Schulte’s A299406 coefficients equal the Liouville function times A210826.

N denotes the natural numbers including zero, Z the integers, and ArithmeticFunction(Z) the integer-valued functions on N that vanish at zero. The indices n and k are natural numbers, and f is such an arithmetic function. The arithmetic function zeta is one on positive inputs and zero at zero; mu is the Mobius function; intCast(zeta) is its integer-valued cast, the coercion from ArithmeticFunction(N) to ArithmeticFunction(Z) that the convolutions with the integer-valued mu require. Omega counts prime factors with multiplicity, and lambda denotes Mathlib’s liouville, A008836: at a positive index it is minus one raised to Omega of that index. A and B denote A299406 and A210826. The binary star is Dirichlet convolution, while the centered dot is integer multiplication of values. Nat.floorRoot divides each prime-factor exponent by k using natural-number division and returns zero when k or n is zero; it is the root for the divisibility order. The function liftPow keeps the value of f at this root exactly when its k-th power equals n. For positive k, its Dirichlet series is obtained by replacing the series variable s by k*s. Reading the Dirichlet generating function coefficientwise sends zeta(6s) to liftPow(6,zeta) and the reciprocals of zeta(2s) and zeta(3s) to liftPow(2,mu) and liftPow(3,mu). In the Lambert series, x is a formal variable: the sum of B(d) over positive divisors d of n equals the cube indicator, so Mobius inversion gives B as mu convolved with that indicator. Only the A299406 formula a(n) = A008836(n) * A210826(n) is asserted here, under these coefficient readings. Analytic convergence and other OEIS assertions are outside the claim. The positive-index hypothesis follows the offset one of A299406.

Definition 1.1 (The exact power lift).

$$\forall k\in \mathbb{N},\forall f\in \mathrm{ArithmeticFunction}\left(\mathbb{Z}\right),\forall n\in \mathbb{N},((\mathrm{floorRoot}{\left(k,n\right)}^k=n)⟹(\mathrm{liftPow}\left(k,f\right)\left(n\right)=f\left(\mathrm{floorRoot}\left(k,n\right)\right)))\wedge ((\mathrm{floorRoot}{\left(k,n\right)}^k\ne n)⟹(\mathrm{liftPow}\left(k,f\right)\left(n\right)=0))$$

Formalization. D5/S3/Arith/SchulteLiouvilleCubeMobius.liftPow (✓ std3).

Source. Repository-derived.

Commentary.

The two defining clauses retain f at the factorization root on exact powers and give zero otherwise. They also describe the total definition when k is zero; the coefficient interpretation uses only positive k. Each displayed zero in a value equality is an integer.

Definition 1.2 (The A299406 coefficient function).

$$A=((\mathrm{intCast}\left(\zeta \right)\ast \mathrm{liftPow}\left(6,\mathrm{intCast}\left(\zeta \right)\right))\ast \mathrm{liftPow}\left(2,\mu \right))\ast \mathrm{liftPow}\left(3,\mu \right)$$

Formalization. D5/S3/Arith/SchulteLiouvilleCubeMobius.A (✓ std3).

Citation. Werner Schulte (2018). OEIS A299406, coefficients of zeta(s) zeta(6s) / (zeta(2s) zeta(3s)), with the Liouville–A210826 product conjecture. URL: https://oeis.org/A299406.

Commentary.

The four convolution factors encode zeta(s) zeta(6s) divided by zeta(2s) zeta(3s). The zeta factors here are integer-valued, matching the natural-to-integer coercions in the Lean definition.

Definition 1.3 (The Lambert-series coefficient function).

$$B=\mu \ast \mathrm{liftPow}\left(3,\mathrm{intCast}\left(\zeta \right)\right)$$

Formalization. D5/S3/Arith/SchulteLiouvilleCubeMobius.B (✓ std3).

Citation. Werner Schulte (2018). OEIS A299406, coefficients of zeta(s) zeta(6s) / (zeta(2s) zeta(3s)), with the Liouville–A210826 product conjecture. URL: https://oeis.org/A299406.

Commentary.

The function liftPow(3,zeta) is one on positive cubes and zero elsewhere. Convolution with mu inverts the positive-divisor sum in the Lambert series defining A210826.

Theorem 1.4 (Schulte’s product formula).

$$\forall n\in \mathbb{N},(0<n)⟹(A\left(n\right)=\lambda \left(n\right)\cdot B\left(n\right))$$

Proof. Machine-checked in Lean as D5/S3/Arith/SchulteLiouvilleCubeMobius.result (✓ std3). ∎

Resolves. Problems/oeis-a299406-schulte-liouville-cube-mobius (proved) by D5/S3/Arith/SchulteLiouvilleCubeMobius.result.

Citation. Werner Schulte (2018). OEIS A299406, coefficients of zeta(s) zeta(6s) / (zeta(2s) zeta(3s)), with the Liouville–A210826 product conjecture. URL: https://oeis.org/A299406.

Commentary.

Every convolution factor is multiplicative. On a prime power with exponent e, both sides have values 1, 1, 0, -1, -1, 0 as e runs through the six residue classes. Equality on all prime powers therefore gives equality at every positive natural index.

References

• Truth anchor: D5/S3/Arith/SchulteLiouvilleCubeMobius.A
• Truth anchor: D5/S3/Arith/SchulteLiouvilleCubeMobius.B
• Truth anchor: D5/S3/Arith/SchulteLiouvilleCubeMobius.liftPow
• Truth anchor: D5/S3/Arith/SchulteLiouvilleCubeMobius.result