Jacobi’s exact four-square formula makes r4(n) elegant, but five squares lead to deeper territory with half-integral weight forms and L-functions. In this episode we trace Emil Grosswald’s clever reduction of r5(n) to a sum of r4(n), bypassing the circle method to yield a sharp asymptotic, and we unpack the main term, the role of L-series, the cusp-form error, and what this reveals about the boundary between exact formulas and structured approximations in number theory.

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