Skip to content

refactor(NumberTheory): golf Mathlib/NumberTheory/ArithmeticFunction/Zeta #38278

New issue

Have a question about this project? Sign up for a free GitHub account to open an issue and contact its maintainers and the community.

Sign up for GitHub

By clicking “Sign up for GitHub”, you agree to our terms of service and privacy statement. We’ll occasionally send you account related emails.

Already on GitHub? Sign in to your account

Open
yuanyi-350 wants to merge 2 commits into
base: master
Choose a base branch
from
+6 −10
Open

refactor(NumberTheory): golf Mathlib/NumberTheory/ArithmeticFunction/Zeta #38278

Changes from all commits
Commits
Show all changes
2 commits
Select commit Hold shift + click to select a range
c4aa6b0
refactor(NumberTheory): golf ArithmeticFunction/Zeta
yuanyi-350 Apr 20, 2026
2455328
apply suggestions
yuanyi-350 Apr 21, 2026
File filter

Filter by extension

Filter by extension
Conversations
Failed to load comments.
Loading
Jump to
Jump to file
Failed to load files.
Loading
Diff view
Diff view
16 changes: 6 additions & 10 deletions Mathlib/NumberTheory/ArithmeticFunction/Zeta.lean
View file
Open in desktop
Original file line number Diff line number Diff line change
Expand Up @@ -74,22 +74,18 @@ theorem sum_divisorsAntidiagonal_eq_sum_divisors {M} [Semiring R] [AddCommMonoid
∑ i ∈ divisors x, f i := by
simp [← coe_zeta_smul_apply (R := R)]

theorem coe_zeta_mul_comm [Semiring R] {f : ArithmeticFunction R} : ζ * f = f * ζ := by
ext x
rw [mul_apply, ← map_swap_divisorsAntidiagonal, Finset.sum_map]
simp [mul_apply]
Comment on lines 77 to +80
Copy link
Copy Markdown
Collaborator

@FLDutchmann FLDutchmann Apr 20, 2026

Choose a reason for hiding this comment

The reason will be displayed to describe this comment to others. Learn more.

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.

Copy link
Copy Markdown
Collaborator

@FLDutchmann FLDutchmann Apr 20, 2026

Choose a reason for hiding this comment

The reason will be displayed to describe this comment to others. Learn more.

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.

Copy link
Copy Markdown
Collaborator Author

@yuanyi-350 yuanyi-350 Apr 21, 2026

Choose a reason for hiding this comment

The reason will be displayed to describe this comment to others. Learn more.

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?


theorem coe_zeta_mul_apply [Semiring R] {f : ArithmeticFunction R} {x : ℕ} :
(ζ * f) x = ∑ i ∈ divisors x, f i :=
coe_zeta_smul_apply

theorem coe_mul_zeta_apply [Semiring R] {f : ArithmeticFunction R} {x : ℕ} :
(f * ζ) x = ∑ i ∈ divisors x, f i := by
rw [mul_apply]
trans ∑ i ∈ divisorsAntidiagonal x, f i.1
· refine sum_congr rfl fun i hi => ?_
rcases mem_divisorsAntidiagonal.1 hi with ⟨rfl, h⟩
rw [natCoe_apply, zeta_apply_ne (right_ne_zero_of_mul h), cast_one, mul_one]
· rw [← map_div_right_divisors, sum_map, Function.Embedding.coeFn_mk]

theorem coe_zeta_mul_comm [Semiring R] {f : ArithmeticFunction R} : ζ * f = f * ζ := by
ext x
rw [coe_zeta_mul_apply, coe_mul_zeta_apply]
rw [← coe_zeta_mul_comm, coe_zeta_mul_apply]

theorem zeta_mul_apply {f : ArithmeticFunction ℕ} {x : ℕ} : (ζ * f) x = ∑ i ∈ divisors x, f i := by
rw [← natCoe_nat ζ, coe_zeta_mul_apply]
Expand Down
Loading