Ok, if z(s)= 1+1/(2^s)+1/(3^s)+1/(4^s)+1/(5^s)+1/(6^s)+1/(7^s)+1/(8^s)+...
then if s=-1 then z(-1)=1+2+3+4+5+6+7+8... now you may know that this in the end all equals -(1/12) but im not interested in that in this entry.
and if s=-2 then z(-2)=1+4+9+46+25+36+49+64
Now the difference between each of these numbers are as follows: 3,5,7,9,11,13,15...
Notice that the difference between THOSE sets of number is this: 2,2,2,2,2,2,2,2....
Interesting. I wonder what would happen if i did z(-3)=1+8+27+64+125+216+343+512....
and Thus that would mean that the difference between those number is: 7,19,37,61,91,2127,169,217....
I notice that the next differentiation is not "3" apart to my disappointment and is: 12,18,24,30,36,42,48,54....
But wait... If i do it one more time i get: 6,6,6,6,6,6,6....
Well I am not really sure what this means at this point but I know it has to have a meaning right?
What about z(-4)=1+16+81+256+625+1296+2401+4096.....
Diff level one: 15 65 175 369 671 1105 1695 2465
Diff level two: 50 110 194 302 434 590 770 974
diff level three: 60 84 108 132 156 180 204 228
Diff level four: 24 24 24 24 24 24 24 24
! I see a pattern emerging. For every whole negative s of the zeta function the s'th difference will be a constant.
I wonder if the constants have a pattern?:
2,6,24,120,720,5040,40320,362880,3628800.....
This does have a recurrence relationship of a(n+1) = (n+2)a(n) when n>1.
You may ask yourself "so what?" and to that I would say... "idk but its interesting"