# A recurance relationship between all zeta's?

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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?

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"
Published:
the sum of the natural numbers diverges, so you shouldn't base much on it being some number
Very true, but it stuck me as interesting.  Not really basing anything on it really.  I have just been trying to understand the Riemann hypothesis (or at least know enough so when it gets proved I might read the proof with a small chance of real understanding. )
'idk but it's interesting' is as excellent a reason as any to explore math and numbers and anything else, really.

here's one of my favourite little clips about Moebius transformations that references Riemann: www.youtube.com/watch?v=0z1fIs…
I agree. It I love getting that fuzzy feeling in my brain when the math does fun things. I dont always understand but that matters not!

That was a cool video.  Fractals are kind of like that, except the extra dimension is the complex plane that your are "looking down from" and the shape is the mandelbrot set.  Kinda. Almost.  Maybe.
right?! I love that so much, too -- when you can feel yourself learning the tiniest bit of some massive concept. it's like hanging onto the edge of a cliff.
i was obsessed with numbers for a time.  in particular, the number 9 has some very interesting properties if you look into it...
Yea, Its fun to think about. Ive been watching videos about he Riemann hypothesis lately.