We have,

For the next step, Renzo Sprugnoli gave the Ramanujan-like identity,

(The sign of the third term has been changed by this author.) However, to make it more symmetrical, we can express the *arctan* in terms of the *log function*. Since,

then,

In this manner, it reduces to the concise,

where, and are the appropriate roots of,

I found that, curiously, *the argument of the log can be expressed in terms of the Dedekind eta function*, . Let,

then,

*Is this coincidence?* Furthermore, using these as the argument of the *polylogarithm*,

one can find a *polylogarithm ladder* to express *Apery’s constant.* For example, getting the square root and reciprocal of so that *z* < 1,

then,

A simpler one exists for the other argument. The next step, of course, is,

Since the first three *Fermat primes* 3, 5, 17 have already appeared, it should be interesting to conjecture if 257 will be next.