The Dedekind eta function $\eta(\tau)$ giev by : $$\eta(\tau)=e^{\frac{\pi i \tau}{12}}\prod_{n=1}^{\infty}\left(1-e^{2\pi i n\tau}\right)\;\;\;\;\;\;\Im(\tau)>0$$ Has the real line as a natural boundary. Suppose that $\tau_{0}$ and $z_{0}$ are two real, irrational numbers. Is there a general method to calculate the limit (if it exists): $$\lim_{(\tau,z)\rightarrow (\tau_{0},z_{0})} \frac{\eta(\tau)}{\eta(z)}$$

EDIT : It's known that for a sufficiently small $y$, we have the asymptotic estimate : $$\left | \eta\left(\left \{ \frac{h}{k} \right \} +iy\right ) \right |_{y\rightarrow 0^{+} }\rightarrow \frac{1}{\sqrt{ky}}e^{-\frac{1}{12k^{2}y}}$$ where $h, k $ are coprime positive integers, and $\left \{ \cdot \right \}$ is the fractional part function. Is there a similar formula for irrational points $\tau=\tau_{0}$ : $$\left | \eta\left(\tau_{0}+iy\right)\right|_{y\rightarrow 0^{+}}$$


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