Sorry, let me pose the full problem and I think you can help me. For any positive integer $x$, let $t(x)$ be a real number with a priori is such that $t(x)>1$ and $t(x)$ tends to $1$ as $x\to \infty$.

Now, denote $q=q(x)=e^{-2\pi t(x)}$ and $q_0=e^{-2\pi}$.

I have that $$ j(q)-j(q_0)=e^{2\pi t(x)}-e^{2\pi}+\sum_{k\geq 1}c_k(e^{-2k\pi t(x)}-e^{-2k\pi}). $$

Thus, all I wish is an explicit lower bound for $j(q)-j(q_0)$ in terms of $t(x)-1$.

Sorry for the previous post. I thank for all suggestions and ideas for bounding $(d_k)$ (which was one of my ideas to get the previous lower bound).

Any hint?

We have the $j$-invariant defined as

I have that $$ j(\tau)=\frac{1}{q}+\sum_{k\geq 0}c_kq^k, $$ where $q=e^{-2\pi t}$ ($\tau=it$) and $c_k\sim e^{4\pi\sqrt{k}}/(k^{3/4}\sqrt{2})$.

The inversion formula for the $j$-invariant is $$ q=j^{-1}+\sum_{k\geq 2}d_kj^{-k}. $$

Thus, I would like to know some upper bound or asymptotic formula for $d_k$.

Any hint or reference?

Sorry, let me pose the full problem and I think you can help me. For any positive integer $x$, let $t(x)$ be a real number with a priori is such that $t(x)>1$ and $t(x)$ tends to $1$ as $x\to \infty$.

Now, denote $q=q(x)=e^{-2\pi t(x)}$ and $q_0=e^{-2\pi}$.

I have that $$ j(q)-j(q_0)=e^{-2\pi t(x)}-e^{-2\pi}+\sum_{k\geq 1}c_k(e^{-2k\pi t(x)}-e^{-2k\pi}). $$

Thus, all I wish is an explicit lower bound for $j(q)-j(q_0)$ in terms of $t(x)-1$.

Sorry for the previous post. I thank for all suggestions and ideas for bounding $(d_k)$ (which was one of my ideas to get the previous lower bound).

Any hint?

Sorry, let me pose the full problem and I think you can help me. For any positive integer $x$, let $t(x)$ be a real number with a priori is such that $t(x)>1$ and $t(x)$ tends to $1$ as $x\to \infty$.

Now, denote $q=q(x)=e^{-2\pi t(x)}$ and $q_0=e^{-2\pi}$.

I have that $$ j(q)-j(q_0)=e^{2\pi t(x)}-e^{2\pi}+\sum_{k\geq 1}c_k(e^{-2k\pi t(x)}-e^{-2k\pi}). $$

Thus, all I wish is an explicit lower bound for $j(q)-j(q_0)$ in terms of $t(x)-1$.

Sorry for the previous post. I thank for all suggestions and ideas for bounding $(d_k)$ (which was one of my ideas to get the previous lower bound).

Any hint?

We know that $q$-expansion of the $j$-invariant function is given by $$ j(\tau)=q^{-1}+744+196884q+21493760q^2+\cdots = q^{-1}+\sum_{k\geq 0}c_kq^k, $$ where $q=e^{2\pi i\tau}$.

Someone know about the existence of an (almost) explicit inverse function of $j$? i.e., such that $$ q=j^{-1}+\sum_{k\geq 2}d_kj^{-k}, $$ where $j=j(\tau)$. I heard about such a formula and a few of initial values of $d_k$. In fact, $d_1=1$, $d_2=750420$, etc.

I would like to find an explicit upper bound for the series $$ \sum_{k\geq 1}\frac{kd_k}{(1728)^{k+1}} $$ in terms of $1/(q-q_0)$, where $q_0=e^{-2\pi}$.

Any hint?

Sorry, let me pose the full problem and I think you can help me. For any positive integer $x$, let $t(x)$ be a real number with a priori is such that $t(x)>1$ and $t(x)$ tends to $1$ as $x\to \infty$.

Now, denote $q=q(x)=e^{-2\pi t(x)}$ and $q_0=e^{-2\pi}$.

I have that $$ j(q)-j(q_0)=e^{-2\pi t(x)}-e^{-2\pi}+\sum_{k\geq 1}c_k(e^{-2k\pi t(x)}-e^{-2k\pi}). $$

Thus, all I wish is an explicit lower bound for $j(q)-j(q_0)$ in terms of $t(x)-1$.

Sorry for the previous post. I thank for all suggestions and ideas for bounding $(d_k)$ (which was one of my ideas to get the previous lower bound).

Any hint?