Troll

A read through the Wolfram article on Riemann's zeta paper
reaasures me that my working formula is indeed the correct
one for the whole complex plane. And I am starting to see how
each computational chunk of it would need a lot of work to see
what is going on.

Looking at the uses of s at either end, they behave in opposition
for both s greater than 1 - the Euler zeta for reals - and for s between
0 and 1, thus fractions and possibly complex expressions. This reflects
Riemanns multiplication of the old zeta with the gamma functions
to form his new zeta. (Is this a troll from the 19th century!?)

*     *     *

Not really complicated numerically, the first term is worth 2

for s at 2, and -5/2 for s at .5. It is a constant which can be

considered at each turn of fraction summation conceptually, but 

stated outside as a mathematical simplification.

The last term referencing s - as a funtion of k - is not a constant,

and will have a different effect with the different number of

iterations we ask for.

It is the work of the first term that shows the zeta is undefined

at 1.

*     *     *

Of more immediate concern, my transcription of the formula on
the (very useful) Desmos. It is clear that the fractions summation -
if we are careful about the zero value - goes to 1. And one is
tempted to dismiss the expression. THAT WOULD BE A SERIOUS
MISTAKE! In point of fact, it is never 1. And since we are looking
for zero coincidences...

The (-1) oscillation produces some zero factors, in its sums.

The binomial expression plays out entirely as a function the
n value we assign.

🙀