The authors show that ∂²/∂x∂y of the theta and Epstein zeta functions is strictly positive, and ∂³/∂x∂y² is strictly negative, in the relevant fundamental domain.
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Signs of high order derivatives for the theta and Epstein zeta functions and application
The authors show that ∂²/∂x∂y of the theta and Epstein zeta functions is strictly positive, and ∂³/∂x∂y² is strictly negative, in the relevant fundamental domain.