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Recall the $j$-invariant function, namely, $$ j(\tau)=\frac{1}{q}+\sum_{k\geq 0}c_kq^k, $$ where $q=e^{2\pi i \tau}$ and the coefficients $(c_k)_k$ are in the OEIS sequence A000521.

By using some normalisation and derivative (I'll omit the details), it is possible to prove that $$ e^{2\pi}=\sum_{k\geq 1}kc_ke^{-2k\pi}. $$ Thus, I would like to study the following problem:

Problem. Let $\epsilon>0$ be a real number, $\alpha_0,\alpha_1,\ldots\in (1,1+\epsilon)$ and set $\delta:=\min_{i}\{\alpha_i\}$. How to obtain an effective lower bound for $$ \left|e^{2\alpha_0\pi}-\sum_{k\geq 1}kc_ke^{-2k\alpha_i\pi}\right| $$ in terms of $\delta-1$?

I believe something like $(\delta-1)^{2+o(1)}$ should work, but I have no success by using any approach.

Could you guys please help me with that?

Many thanks in advance!

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  • $\begingroup$ I don't see how that would work - you can let all $\alpha_i\to 1$ while keeping them in $(1,1+\epsilon)$, and the difference tends to $0$. $\endgroup$
    – Wojowu
    Commented Dec 21, 2021 at 23:03
  • $\begingroup$ You are right. I made some confusion. I just edited, thanks! $\endgroup$
    – Jean
    Commented Dec 22, 2021 at 3:22

1 Answer 1

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$\frac{j'(i+z)}{z}$ is analytic and non-zero on $|z|<1/100$, it attains its minimum and maximum modulus on $|z|=1/100$.

Therefore $$\left|\frac{j'(i+z)}{z}\right| \in (48000,50000)$$

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  • $\begingroup$ Thank you very much for your answer. Could you please provide me some reference? Also, it is possible to use the same idea do obtain a lower bound for $e^{2\pi t}-e^{2\pi}+\sum_{k\geq 1}c_k(e^{-2k\pi t}-e^{-2k\pi})$ in function of $(t-1)$? $\endgroup$
    – Jean
    Commented Dec 22, 2021 at 14:09
  • $\begingroup$ A reference for what? $\endgroup$
    – reuns
    Commented Dec 22, 2021 at 14:14
  • $\begingroup$ For $j'(i+z)/z$ not zero on that open disc. But I think it is ok. $\endgroup$
    – Jean
    Commented Dec 22, 2021 at 14:37
  • $\begingroup$ $j(a)=j(b)$ iff $b\in SL_2(\Bbb{Z})a$ gives that $j'(\tau)=0$ iff $\tau\in SL_2(\Bbb{Z})i \cup SL_2(\Bbb{Z})e^{2i\pi/3}$ and $j''(i)\ne 0$. So it remains to check that $|\tau-i|<1/100$ doesn't contain any of those, pretty obvious from this picture of the $SL_2(\Bbb{Z})$-orbit of the standard fundamental domain in the upper half-plane $\endgroup$
    – reuns
    Commented Dec 22, 2021 at 14:41
  • $\begingroup$ Thanks! But how the minimum and maximum of $|j'(i+z)/z|$ can be applied to my problem? Sorry, I didn't realize it. $\endgroup$
    – Jean
    Commented Dec 22, 2021 at 16:17

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