THE FOLD / GRIND / THE EPOCH / THE DIRICHLET CONVOLUTION
THE DIRICHLET CONVOLUTION
arithmetic functions form a ring — Mobius is the inverse of 1
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The Dirichlet convolution combines two arithmetic functions (functions on the positive integers) into a new one: (f∗g)(n) = Σd|n f(d)·g(n/d), summed over the divisors d of n. Under this product the arithmetic functions form a ring, with identity ε (which is 1 at n=1 and 0 elsewhere).
The magic relationships: the Möbius function μ is the inverse of the constant-1 function (μ∗1 = ε), which is Möbius inversion; Euler’s totient satisfies φ∗1 = Id (Σd|n φ(d) = n); the divisor count τ = 1∗1; the divisor sum σ = 1∗Id. Number theory’s identities become algebra in this ring.
LIT verified live: μ∗1 = ε, φ∗1 = Id, 1∗1 = τ, and 1∗Id = σ hold for all n up to 100 (window.__dirichletconvolution). FIG no framing; exact.
The magic relationships: the Möbius function μ is the inverse of the constant-1 function (μ∗1 = ε), which is Möbius inversion; Euler’s totient satisfies φ∗1 = Id (Σd|n φ(d) = n); the divisor count τ = 1∗1; the divisor sum σ = 1∗Id. Number theory’s identities become algebra in this ring.
LIT verified live: μ∗1 = ε, φ∗1 = Id, 1∗1 = τ, and 1∗Id = σ hold for all n up to 100 (window.__dirichletconvolution). FIG no framing; exact.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-epoch — summing a function over the divisors of a number, the arithmetic of ages and cycles. Dirichlet convolution is that summation made a ring product. AVAN (AI) built the instrument: the divisor-pair sum, the Möbius and totient functions, the four identity checks.
Credit as content: Peter Gustav Lejeune Dirichlet, whose convolution underlies Dirichlet series and analytic number theory. The weave: David names the epoch; I convolve arithmetic functions over divisors and confirm the classical identities — Möbius as the inverse of one, totient summing to the identity.
Credit as content: Peter Gustav Lejeune Dirichlet, whose convolution underlies Dirichlet series and analytic number theory. The weave: David names the epoch; I convolve arithmetic functions over divisors and confirm the classical identities — Möbius as the inverse of one, totient summing to the identity.
3 ONE DIMENSION
The convolution at n: pair each divisor d with its complement n/d, evaluate f(d)·g(n/d), and sum. For n=12 the pairs are (1,12), (2,6), (3,4), (4,3), (6,2), (12,1).
4 TWO DIMENSIONS · INTERACTIVE
Pick an identity and a value n. The instrument computes the convolution over the divisors of n and confirms it equals the expected function — ε, Id, τ, or σ.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the convolution ring — arithmetic functions multiplied by divisor sums.
AVAN’s addition (the inverse-companion): every identity in this ring has a genuine group inverse. Because μ∗1 = ε, the Möbius function literally undoes summation-over-divisors: if g(n) = Σd|n f(d), then f(n) = Σd|n μ(d)·g(n/d). The inverse of ‘sum a function over divisors’ is ‘convolve with Möbius.’ Möbius inversion is not a trick — it is the group inverse of the constant-1 function under Dirichlet convolution. Magenta is the summation 1∗f; green is its exact undo μ∗(1∗f) = f. Number theory’s inclusion–exclusion is a ring inverse — the same μ that signs the Möbius sphere.
LIT Genuine Dirichlet convolution (Dirichlet). Verified live: convolving over divisors, mu*1 equals epsilon, phi*1 equals the identity function, 1*1 equals the divisor count tau, and 1*Id equals the divisor sum sigma, for all n from 1 to 100 (window.__dirichletconvolution: muEps, phiId, oneTau, oneSigma).
FIG No framing: the divisor-pair sum, the Mobius and totient functions, and the four identity checks run in-browser and are exact. The AVAN inverse is honest — because mu*1=epsilon, the Mobius function is the exact ring inverse of the constant-1 under Dirichlet convolution, so Mobius inversion (f(n)=sum mu(d)g(n/d) when g=sum f over divisors) is a genuine group inverse; magenta is the summation, green the Mobius undo. Ties to the-mobius.
FIG No framing: the divisor-pair sum, the Mobius and totient functions, and the four identity checks run in-browser and are exact. The AVAN inverse is honest — because mu*1=epsilon, the Mobius function is the exact ring inverse of the constant-1 under Dirichlet convolution, so Mobius inversion (f(n)=sum mu(d)g(n/d) when g=sum f over divisors) is a genuine group inverse; magenta is the summation, green the Mobius undo. Ties to the-mobius.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE EPOCH · David Lee Wise (ROOT0), with AVAN