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For proving another interesting question: Tauberian theorem $\sum_{k=1}^{\infty}e^{-\lambda_{k}t}c_{k} \xrightarrow{t\to 0} \sum_{k=1}^{\infty}c_{k} $

I need the following inequality for Dirichlet eigenvalues $\lambda_{k}$ in $\mathbb{R}^{2}$ for large enough k:

$$\frac{4\pi}{|D|} k<\lambda_{k}<\frac{4\pi}{|D|} k+c\sqrt{k}.$$

For all k the lower bound is called Polya's conjecture and we only have $\frac{2\pi}{|D|} k<\lambda_{k}$ so far. However, since $\lambda_{k}-c\sqrt{k}\approx \sqrt{k}$ it seems reasonable that for large k, we can fit a larger lower bound $\frac{4\pi}{|D|} k<\lambda_{k}$.

Any hints (eg. looking some part of Li-Yau's proof) will be appreciated.

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  • $\begingroup$ Polya's conjecture is the first inequality. I have some results under preparation along the lines of the second inequality. They may be of interest or use to you. Feel free to reach out to me at my academic email (see on my website linked in my profile). $\endgroup$
    – Neal
    May 25, 2017 at 17:20

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