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I need help with this one from Stein & Shakarchi.

Let $$E_{4}(\tau)=\sum_{(n,m)\neq (0,0)}{\frac{1}{(n+m\tau)^{4}}}$$ be the Eisenstein series of order $4$. Show that $$\left|E_{4}(\tau)-\frac{\pi^{4}}{45}\right|\leq c e^{-2\pi t}\ \text{if}\ \tau = x + it\ \text{and}\ t \geq 1.$$ and deduce that $$\left|E_{4}(\tau)-\tau^{-4}\frac{\pi^{4}}{45}\right|\leq c t^{-4} e^{-2\pi/ t}\ \text{if}\ \tau = it\ \text{and}\ 0 < t \leq 1.$$

My idea is to use the estimates from the proof of Theorem 2.5 of pp. 277, but I can't see a finish.

Thanks in advance!

EDIT: On page 277, we have the information that in general $$E_{k}(\tau)=\sum_{(n,m)\neq (0,0)}{\frac{1}{(n+m\tau)^{k}}} = 2 \zeta(k) + \frac{2 (-1)^{k/2} (2 \pi)^{k} }{(k-1)!} \sum_{m >0 } {\sum_{l=1}^{\infty}{l^{k-1}e^{2\pi i \tau m l }}}.$$

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You might better your chances of getting help if you mention what the estimates on pg. 277 are. –  Dylan Moreland Dec 11 '11 at 20:11
    
Edited, thanks. –  Anna Dec 11 '11 at 23:47
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1 Answer 1

up vote 2 down vote accepted

This was not so bad as I thought. I actually managed to do it by using the crude estimate $\sigma_{3}(r) \leq r^{4} \leq e ^{2\pi t r}$ for $t \geq 1$ (the first part); as for the second part, it is just a matter of tricking it using the first estimate using the inversion formula for $E_{k}(\tau)$.

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