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We prove some differential equations for the Riemann theta functionassociated to the Jacobian of a Riemann surface. The proof is based on somevariants of a formula by Fay for the theta function, which are motivated bytheir analogues in Arakelov theory of Riemann surfaces. Moreover, we give ageneralization of Rosenhain's formula to hyperelliptic Riemann surfaces asconjectured by Gu\`ardia.

Some differential equations for the Riemann $\theta$-function on Jacobians