Given ,
1. ,
2. ,
3. ,
where .
I have tried to write a program to compute the Zeta function for complex arguments. In particular, I wanted to look at non trivial zeros.
My HP calc and my program agree on some spot checked terms.
For example, 1 / (2 ^ s) where s is equal to (.5 + 14.1347), my program and my HP calculator agree on the answer.
But after adding the first 1000 terms, I don't see any obvious convergence.
How fast does the Zeta function converge? Can you think of any obvious way for me to have made mistakes when adding the terms, or mistakes in my understanding?
Your efforts are admirable... but I suppose you have neglect a little minor particular...
... the particular you have neglect is that the expression...
(1)
... holds only for . If you instruct your HP calculator to compute (1) for may be you don't find as result ...
Kind regards
The forumula you have written is nothing else that this...
(1)
... that was 'discovered' by Leonhard Euler in XVIII° century and is commonly known as 'Reflection relation'. Effectively once you know and for a given s, you automatically know . The trouble is that a formula like this...
(2)
... holds only for , so that (1) and (2) are useful to you only to arrive to compute for . If I undestand correctly you are searching the zeroes of in the 'critical strip', where is ...
It is evident that the problem is not so easy to approach... anyway... never say never again! ...
Kind regards