[Merged by Bors] - refactor(NumberTheory): golf Mathlib/NumberTheory/ArithmeticFunction/Zeta

[yuanyi-350]

• proves coe_zeta_mul_comm directly by reindexing the divisor antidiagonal with map_swap_divisorsAntidiagonal
• derives coe_mul_zeta_apply from coe_zeta_mul_comm and coe_zeta_mul_apply

Extracted from #38144

[github-actions]

PR summary 4fceb4697e

Import changes for modified files

No significant changes to the import graph

Import changes for all files

Files	Import difference

---

Declarations diff

No declarations were harmed in the making of this PR! 🐙

You can run this locally as follows

## summary with just the declaration names:
./scripts/pr_summary/declarations_diff.sh <optional_commit>

## more verbose report:
./scripts/pr_summary/declarations_diff.sh long <optional_commit>

The doc-module for scripts/pr_summary/declarations_diff.sh contains some details about this script.

---

No changes to technical debt.

This script lives in the mathlib-ci repository. To run it locally, from your mathlib4 directory:

git clone https://github.com/leanprover-community/mathlib-ci.git ../mathlib-ci
../mathlib-ci/scripts/reporting/technical-debt-metrics.sh pr_summary

• The relative value is the weighted sum of the differences with weight given by the inverse of the current value of the statistic.
• The absolute value is the relative value divided by the total sum of the inverses of the current values (i.e. the weighted average of the differences).

[yuanyi-350]

I ran a profiler comparison for the changed declarations in this PR.

Results (seconds):

• ArithmeticFunction.coe_zeta_mul_comm: 0.027800 -> 0.108736 (+291.1%)
• ArithmeticFunction.coe_mul_zeta_apply: 0.031816 -> 0.036930 (+16.1%)

Overall:

• Both declarations remain small in absolute terms; coe_zeta_mul_comm is slower by about 0.081s, while coe_mul_zeta_apply is roughly unchanged.

[yuanyi-350]

!radar

[leanprover-radar]

Benchmark results for c4aa6b0 against 4fceb46 are in. No significant results found. @yuanyi-350

• ✅ build//instructions: -7.3G (-0.00%)

Small changes (1✅, 1🟥)

• ✅ build/module/Mathlib.Tactic.Basic//instructions: -290.4M (-4.16%)
• 🟥 build/module/Mathlib.Tactic.Linarith//instructions: +270.6M (+6.33%)

[FLDutchmann]

Thanks! These proofs are much nicer.

[FLDutchmann]

Can you move this to be above coe_zeta_mul_apply? That way moe_coe_zeta and coe_zeta_mul will be next to eachother in the docs.

[FLDutchmann]

Also this lemma is true for zeta replaced by any ArithmeticFunction Nat, but if this breaks anything I'm happy to fix this in a follow-up PR.

[yuanyi-350]

Thank you for the suggestion! I'm actually not an expert in this field and don't completely understand the broader definitions. Given that, I'd rather not touch this part right now just to be safe. Would it be okay if we hold off on this change for now?

[MichaelStollBayreuth]

maintainer merge

[github-actions]

🚀 Pull request has been placed on the maintainer queue by MichaelStollBayreuth.

[riccardobrasca]

Thanks!

bors merge

[mathlib-bors]

Pull request successfully merged into master.

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