Alphabeta Math
ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-31
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A small lattice decomposition for Dirichlet's hyperbola method

Example

Take x=12, U=3, and V=4, so UV=12. For τ=1∗1, Dirichlet's hyperbola identity becomes

∑n≤12τ(n)=∑a≤3⌊12a⌋+∑b≤4⌊12b⌋−3⋅4.

Numerically this is

35=(12+6+4)+(12+6+4+3)−12.

Facts & Assumptions

Given: The value x=12 and the split U=3, V=4.

Verification

technique · direct
1.1

By The divisor functions arise by Dirichlet convolution, τ=1∗1. Therefore Dirichlet's hyperbola method for summatory convolutions applies with F(y)=G(y)=⌊y⌋.

given
2.1

The two arm sums are ∑a≤3⌊12a⌋=12+6+4=22,∑b≤4⌊12b⌋=12+6+4+3=25, and the overlap is F(3)G(4)=3⋅4=12.

step 1.1givenalgebra
3.1

Hence the hyperbola identity gives ∑n≤12τ(n)=22+25−12=35. This matches the direct total 1+2+2+3+2+4+2+4+3+4+2+6=35.

step 2.1givenalgebra∎

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