Granvik’s Möbius and Mertens identities

Abstract

The diagonal and first row of the Granvik Mertens array.

Definition 1.1 (Somos’s rational array).

$$\forall n:\mathbb{N},\forall k:\mathbb{N},T\left(n,k\right)\in \mathbb{Q}$$

Formalization. D5/S3/Arith/GranvikMertensArray.T (✓ std3).

Citation. Mats Granvik; Olivier Gérard; Michael Somos (2011). OEIS A192763 and the Granvik Mertens array. URL: https://oeis.org/A192763.

Commentary.

Indices are natural numbers; a cell with either index zero is zero. On positive indices, T(1,1)=1, T(2,1)=2, and symmetry gives T(n,k)=T(k,n) when k>n. For n at least three, the border update is T(n,1)=(-T(n-1,1)-sum over 2<=i<=n-1 of T(n,i))/(n+1)+T(n-1,1). For n>k>=2, T(n,k)=T(k,((n-1) mod k)+1). For n>=2, T(n,n) is the negative sum of T(n,i) over 1<=i<=n-1. All sums are finite and division is rational. A well-founded measure orders these recursive calls. These are Michael Somos’s Mathematica and PARI programs in OEIS A192763, with their corner normalization one.

Theorem 1.2 (The diagonal and the Mertens row).

$$(\forall n:\mathbb{N},(1\le n)⟹T\left(n,n\right)=n\cdot \mathrm{mu}\left(n\right))\wedge (\forall k:\mathbb{N},(2\le k)⟹T\left(1,k\right)=2+\sum _{j\in \mathrm{Icc}\left(1,k\right)}\mathrm{mu}\left(j\right))$$

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

Resolves. Problems/oeis-a192763-granvik-mertens-array (proved) by D5/S3/Arith/GranvikMertensArray.result.

Source. Repository-derived.

Acknowledgement. Mats Granvik; Olivier Gérard; Michael Somos (2011). OEIS A192763 and the Granvik Mertens array. URL: https://oeis.org/A192763.

Commentary.

Here mu is ArithmeticFunction.moebius, cast from integers to rationals. The first assertion includes the corner n=1; the second begins at k=2, exactly as in the two COMMENTS conjectures in OEIS A192763.

The proof constructs a symmetric model. A noncoprime interior cell has value gcd(n,k)*mu(gcd(n,k)); a coprime interior cell has the border value at the last Euclidean quotient. Signed Bézout identities identify that quotient with floor((n-1)/min(u,n-u)), where u is the inverse residue of k modulo n. Inversion and reflection permute the reduced residues. Möbius convolution evaluates their summed border values, and a gcd multiplicity identity supplies the other cells. The resulting row sum gives the required diagonal and border equations.

The proof also gives a literal definition of the first Mathematica program, attributed to Mats Granvik and Olivier Gérard, including its reverse-window sums. Periodicity and the zero diagonal row sum show that Somos’s array satisfies this program. Well-founded uniqueness identifies both arrays and the arithmetic model. The companion Math.SE 50719 question prints a different corner; the theorem uses the OEIS programs’ value one. No Mertens growth estimate follows from these identities.

References

• Truth anchor: D5/S3/Arith/GranvikMertensArray.T
• Truth anchor: D5/S3/Arith/GranvikMertensArray.result
• Dependency: D5/S3/Arith/RoldanCirculantEnergyP2Q3