feat(NumberTheory): add bernoulli'_five, bernoulli'_six and riemannZeta_six

[qdiazblanco]

Add bernoulli'_five and bernoulli'_six as simp lemmas, continuing the
existing sequence of explicit values bernoulli'_zero through
bernoulli'_four, and use the latter to prove riemannZeta 6 = π ^ 6 / 945.

Along the way, golf the proofs of bernoulli'_two, bernoulli'_three and
bernoulli'_four .

These results are upstreamed from the FLT project (ImperialCollegeLondon/FLT#1069).

Co-authored-by: William Coram
Co-authored-by: Samuel Yin
Co-authored-by: Pepa Montero
Co-authored-by: Archie Browne

---

Original PR by William Coram and Samuel Yin, written with the assistance of Claude and
cleaned up by Codex and then by Pepa Montero).

I have done some further changes to fit Mathlib conventions and Claude was also used to help shorten the
proof of riemannZeta_six with an idea by Archie Browne.

[github-actions]

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[github-actions]

PR summary 886441ec86

Import changes for modified files

No significant changes to the import graph

Import changes for all files

Files	Import difference

---

Declarations diff (regex)

+ bernoulli'_five
+ bernoulli'_six
+ riemannZeta_six

You can run this locally as follows

## from your `mathlib4` directory:
git clone https://github.com/leanprover-community/mathlib-ci.git ../mathlib-ci

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

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

The doc-module for scripts/pr_summary/declarations_diff.sh in the mathlib-ci repository contains some details about this script.

Declarations diff (Lean)

✅ Lean-aware diff — post-build, computed from the Lean environment (commit 886441e).

• +3 new declarations
• −0 removed declarations

+bernoulli'_five
+bernoulli'_six
+riemannZeta_six

---

No changes to strong technical debt.

No changes to weak technical debt.

Current commit 886441ec86
Reference commit ed0d70fdcd

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).

[grunweg]

Thanks for the PR! Let me ask the same question as Yael on the other PR: do we want to have these lemmas, or should we invest in a better tactic instead?

[qdiazblanco]

Thanks for the PR! Let me ask the same question as Yael on the other PR: do we want to have these lemmas, or should we invest in a better tactic instead?

I think you're both right and Bhavik said he's working on it in a comment on the PR I closed yesterday:

I am working on a metaprogram to compute these (which outperforms the methods in that thread), keep an eye out for a PR about this, or feel free to DM me if you need something sooner.

That's certainly the better option, we just need this values for the FLT. So maybe for now it is useful having them?

[grunweg]

If Bhavik is indeed working on them now, I'd rather wait for Bhavik for a moment. The code is already fine in FLT.