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I now realize my question had two parts: 1) asking why the relation $a\approx b$ is not arbitrary; 2) asking what $4\pi$ has to do with black hole entropy.

I just figured out the first part. Why emphasize,

$$\log(196883) \approx 4\pi\tag1$$

when (for example) ,

$$\log(196883) \approx 12\,\zeta(5)$$

is a better fit? The answer was in front of me all long (so I'm a bit embarrassed). All we have do to is use Rademacher's asymptotic formula,

$$c_n \approx \frac{e^{4\pi\sqrt{n}}}{\sqrt{2}\,n^{3/4}}\tag2$$

for the coefficients $c_n$ of the j-function,

$$j(q)-744 = q^{-1}+\sum_{n=1}^\infty c_n q^n$$

Taking the $\log$ of $(2)$, then,

$$\log(c_n) \approx 4\pi\sqrt{n}-\tfrac{3}{4}\log(n)-\tfrac{1}{2}\log(2)\tag3$$

So for $n=1$,

$$\log(196884) \approx 4\pi-\tfrac{1}{2}\log(2)$$

from which $(1)$ necessarily follows.

P.S. I posted the original, short version of my question before I read Witten's paper. (It has 83 pages of math, so that's my excuse.) If I read up to page 34, then $(3)$ is there. I'm more familiar with it expressed as $(2)$ in another question I asked before, so I'm doubly embarrassed. Mea culpa.

However, I still don't know what the heck $4\pi\sqrt{k}$ has to do with black hole entropy.

I now realize my question had two parts: 1) asking why the relation $a\approx b$ is not arbitrary; 2) asking what $4\pi$ has to do with black hole entropy.

I just figured out the first part. Why emphasize,

$$\log(196883) \approx 4\pi\tag1$$

when (for example) ,

$$\log(196883) \approx 12\,\zeta(5)$$

is a better fit? The answer was in front of me all long (so I'm a bit embarrassed). All we have do to is use Rademacher's asymptotic formula,

$$c_n \approx \frac{e^{4\pi\sqrt{n}}}{\sqrt{2}\,n^{3/4}}\tag2$$

for the coefficients $c_n$ of the j-function,

$$j(q)-744 = q^{-1}+\sum_{n=1}^\infty c_n q^n$$

Taking the $\log$ of $(2)$, then,

$$\log(c_n) \approx 4\pi\sqrt{n}-\tfrac{3}{4}\log(n)-\tfrac{1}{2}\log(2)\tag3$$

So for $n=1$,

$$\log(196884) \approx 4\pi-\tfrac{1}{2}\log(2)$$

from which $(1)$ necessarily follows.

P.S. I posted the original, short version of my question before I read Witten's paper. (It has 83 pages of math, so that's my excuse.) If I read up to page 34, then $(3)$ is there. I'm more familiar with it expressed as $(2)$ in another question I asked before, so I'm doubly embarrassed. Mea culpa.

However, I still don't know what the heck $4\pi\sqrt{k}$ has to do with black hole entropy.

(Too long for a comment)

To clarify the question, I now realize it had two parts: 1) asking why the relation $a\approx b$ is not arbitrary; 2) asking what $4\pi$ has to do with black hole entropy.

I just figured out the first part. Why emphasize,

$$\log(196883) \approx 4\pi\tag1$$

when (for example) ,

$$\log(196883) \approx 12\,\zeta(5)$$

is a better fit? The answer was in front of me all long (so I'm a bit embarrassed). All we have do to is use Rademacher's asymptotic formula,

$$c_n \approx \frac{e^{4\pi\sqrt{n}}}{\sqrt{2}\,n^{3/4}}\tag2$$

for the coefficients $c_n$ of the j-function,

$$j(q)-744 = q^{-1}+\sum_{n=1}^\infty c_n q^n$$

Taking the $\log$ of $(2)$, then,

$$\log(c_n) \approx 4\pi\sqrt{n}-\tfrac{3}{4}\log(n)-\tfrac{1}{2}\log(2)\tag3$$

So for $n=1$,

$$\log(196884) \approx 4\pi-\tfrac{1}{2}\log(2)$$

from which $(1)$ necessarily follows.

P.S. I posted the original, short version of my question before I read Witten's paper. (It has 83 pages of math, so that's my excuse.) If I read up to page 34, then $(3)$ is there. I'm more familiar with it expressed as $(2)$ in another question I asked before, so I'm doubly embarrassed. Mea culpa.

However, I still don't know what the heck $4\pi\sqrt{k}$ has to do with black hole entropy.

I now realize my question had two parts: 1) asking why the relation $a\approx b$ is not arbitrary; 2) asking what $4\pi$ has to do with black hole entropy.

I just figured out the first part. Why emphasize,

$$\log(196883) \approx 4\pi\tag1$$

when (for example) ,

$$\log(196883) \approx 12\,\zeta(5)$$

is a better fit? The answer was in front of me all long (so I'm a bit embarrassed). All we have do to is use Rademacher's asymptotic formula,

$$c_n \approx \frac{e^{4\pi\sqrt{n}}}{\sqrt{2}\,n^{3/4}}\tag2$$

for the coefficients $c_n$ of the j-function,

$$j(q)-744 = q^{-1}+\sum_{n=1}^\infty c_n q^n$$

Taking the $\log$ of $(2)$, then,

$$\log(c_n) \approx 4\pi\sqrt{n}-\tfrac{3}{4}\log(n)-\tfrac{1}{2}\log(2)\tag3$$

So for $n=1$,

$$\log(196884) \approx 4\pi-\tfrac{1}{2}\log(2)$$

from which $(1)$ necessarily follows.

P.S. I posted the original, short version of my question before I read Witten's paper. (It has 83 pages of math, so that's my excuse.) If I read up to page 34, then $(3)$ is there. I'm more familiar with it expressed as $(2)$ in another question I asked before, so I'm doubly embarrassed. Mea culpa.

However, I still don't know what the heck $4\pi\sqrt{k}$ has to do with black hole entropy.

(Too long for a comment)

To clarify the question, I now realize it had two parts: 1) asking why the relation $a\approx b$ is not arbitrary; 2) asking what $4\pi$ has to do with black hole entropy.

I just figured out the first part. Why emphasize,

$$\log(196883) \approx 4\pi\tag1$$

when (for example) ,

$$\log(196883) \approx 12\,\zeta(5)$$

is a better fit? The answer was in front of me all long (so I'm a bit embarrassed). All we have do to is use Rademacher's asymptotic formula,

$$c_n \approx \frac{e^{4\pi\sqrt{n}}}{\sqrt{2}\,n^{3/4}}\tag2$$

for the coefficients $c_n$ of the j-function,

$$j(q)-744 = q^{-1}+\sum_{n=1}^\infty c_n q^n$$

Taking the $\log$ of $(2)$, then,

$$\log(c_n) = 4\pi\sqrt{n}-\tfrac{3}{4}\log(n)-\tfrac{1}{2}\log(2)\tag3$$

So for $n=1$,

$$\log(196884) \approx 4\pi-\tfrac{1}{2}\log(2)$$

from which $(1)$ necessarily follows.

P.S. I posted the original, short version of my question before I read Witten's paper. (It has 83 pages of math, so that's my excuse.) If I read up to page 34, then $(3)$ is there. I'm more familiar with it expressed as $(2)$ in another question I asked before, so I'm doubly embarrassed. Mea culpa.

However, I still don't know what the heck $4\pi\sqrt{k}$ has to do with black hole entropy.

(Too long for a comment)

To clarify the question, I now realize it had two parts: 1) asking why the relation $a\approx b$ is not arbitrary; 2) asking what $4\pi$ has to do with black hole entropy.

I just figured out the first part. Why emphasize,

$$\log(196883) \approx 4\pi\tag1$$

when (for example) ,

$$\log(196883) \approx 12\,\zeta(5)$$

is a better fit? The answer was in front of me all long (so I'm a bit embarrassed). All we have do to is use Rademacher's asymptotic formula,

$$c_n \approx \frac{e^{4\pi\sqrt{n}}}{\sqrt{2}\,n^{3/4}}\tag2$$

for the coefficients $c_n$ of the j-function,

$$j(q)-744 = q^{-1}+\sum_{n=1}^\infty c_n q^n$$

Taking the $\log$ of $(2)$, then,

$$\log(c_n) \approx 4\pi\sqrt{n}-\tfrac{3}{4}\log(n)-\tfrac{1}{2}\log(2)\tag3$$

So for $n=1$,

$$\log(196884) \approx 4\pi-\tfrac{1}{2}\log(2)$$

from which $(1)$ necessarily follows.

P.S. I posted the original, short version of my question before I read Witten's paper. (It has 83 pages of math, so that's my excuse.) If I read up to page 34, then $(3)$ is there. I'm more familiar with it expressed as $(2)$ in another question I asked before, so I'm doubly embarrassed. Mea culpa.

However, I still don't know what the heck $4\pi\sqrt{k}$ has to do with black hole entropy.