Mordell’s Proof of Euler Product for the Tau Function

Overview

A concise walkthrough of Mordell’s proof that the Ramanujan tau function $au(n)$ yields an Euler product for the $L$-series of the modular discriminant $\Delta$. The animation builds the generating‑function identity step‑by‑step, ending with the explicit Euler factor.

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Phases

#	Phase Name	Duration	Description
1	Title	~2 s	Fade‑in the title "Mordell’s Proof of Euler Product (Tau Function)" then fade out.
2	Tau Introduction	~2 s	Write the definition "$\ au(n)$ : Ramanujan's tau function" and fade it out.
3	Hecke Relation	~2 s	Display the Hecke operator relation $T(p)\Delta(x)=\ au(p)\Delta(x)$ and fade out.
4	Recurrence Relations	~3 s	Show the two recurrence formulas for $\ au(p^k)$ stacked vertically, hold, then fade out.
5	Goal $L$-Series	~3 s	Present the target Euler product $L(s,\Delta)=\prod_{p}\bigl(\sum_{k=0}^{\infty}\frac{\ au(p^k)}{p^{ks}}\bigr)$ and fade out.
6	Generating Function Setup	~3 s	Write $F_{p}(x)=\sum_{k=0}^{\infty}\ au(p^k)x^{k}$, move it to the top edge.
7	Key Identity	~3 s	Introduce $(1-\ au(p)x+p^{11}x^{2})F_{p}(x)=1$, highlight with a rectangle, then remove the highlight.
8	Solve for $F_{p}(x)$	~3 s	Transform the identity into $F_{p}(x)=\frac{1}{1-\ au(p)x+p^{11}x^{2}}$ and fade both expressions out.
9	Substitution $x=p^{-s}$	~2 s	Write the substitution $x=p^{-s}$.
10	Final Euler Factor	~4 s	Transform the substitution into the closed form $\sum_{k=0}^{\infty}\frac{\ au(p^k)}{p^{ks}}=\frac{1}{1-\ au(p)p^{-s}+p^{11-2s}}$.
11	Closing	~2 s	Show "Euler Product Achieved!" then fade out everything.

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Layout

┌─────────────────────────────────────────────┐
│                 TOP AREA                    │
├─────────────────────────────────────────────┤
│                 MAIN AREA                    │
│   (central region where all equations appear)│
├─────────────────────────────────────────────┤
│                BOTTOM AREA                  │
└─────────────────────────────────────────────┘

Area Descriptions

Area	Content	Notes
Top	Optional short label (e.g., "Mordell Proof") – fades in with the title phase.	Appears only in Phase 1.
Main	All mathematical objects: definitions, relations, generating function, identities, final Euler factor.	Centered; each object replaces the previous one unless explicitly kept (e.g., generating function moved to top).
Bottom	Small caption or source note (e.g., "Source: Mordell 1917") – can stay faint throughout.	Optional; does not interfere with main visual.

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Notes

• The entire animation fits within a single Scene class.
• Text is used only for the title and closing message; all substantive information is conveyed via mathematical expressions.
• Fade‑in/out and simple transforms are preferred to keep the total runtime under 30 seconds (≈31 s with generous pauses; durations can be trimmed if needed).
• The rectangle highlight in Phase 7 should appear briefly to draw attention to the cancellation idea, then disappear.
• No additional background graphics or decorative elements are required.