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.

You can run this locally as

./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%)