Alphabeta MathHow statement and proof provenance work
The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
- Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
- AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
- AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.
These labels describe origin, not correctness: citations and verification chips remain separate evidence.
The Dirichlet-convolution identity and the constant-one function
Definition
The Dirichlet-convolution identity is the arithmetic function given by
The constant-one function is the arithmetic function defined by
for every positive integer .
Remarks
- The symbols and serve different roles. Later identities such as and use both at once.
Depends on
Used by
- A Dirichlet-convolution table through 12 Example
- Computing a Dirichlet inverse recursively Example
- The convolution 1*λ detects perfect squares Proposition
- The divisor functions arise by Dirichlet convolution Proposition
- An arithmetic function has a Dirichlet inverse exactly when its value at 1 is nonzero Theorem
- Arithmetic functions form a commutative ring under pointwise addition and Dirichlet convolution Theorem
Dependency tree · two levels
3 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- Victor Shoup, A Computational Introduction to Number Theory and Algebra, Section 2.9 (standard reference, not scraped)
- Kiran S. Kedlaya, An Introduction to Analytic Number Theory, Definition 3.8 (standard reference, not scraped)