Binary partition at oo ?

[tommy1729]

\(
\frac{\sum_{k=-\infty}^{\infty} \left((-1)^k\, 2^{-k^2/2}\right)}{\sum_{k=-\infty}^{\infty} \left(2^{-k^2/2}\right)}
\)

This evaluates to approximately (0.004872868560797)/(3.01076739115959), which is approximately 0.001618480582427.

In general \( \sum_{k=-\infty}^{\infty} \left((-1)^k\, f(-k^2)\right) \) equals \( 2 \sum_{k=1}^{\infty} \left((-1)^k\, f(-k^2)\right)-f(0) \).

You probably already know that , but maybe it simplifies matters ?

Are you claiming that there are no zero's off the real line ?

Very intresting stuff.

I think we are onto a general thing ;

No zero's in the upper complex plane
Good entire approximations ( fake function , J(x) for the binary p )
... with all derivatives positive.

It all seems connected.

Id like to see how you arrived at these things.

Thanks.


regards

tommy1729