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What are some interesting values of zeta(ni)?
n is an ordinary integer (positive or negative) and i is the imaginary unit.
In case you're wondering, this isn't a homework question. I'm reading a book about the Riemann hypothesis and I looked at the Mathworld article (a lot of the latter is way over my head) and I was just wondering.
2 Answers
- Anonymous1 decade agoFavorite Answer
There might be interesting values, but these have not been researched as much as those for zeta(n). But of course most of the attention has been focused on zeta(1/2 + xi).
This much I can tell you: the value of zeta(ni) is exactly the same as that of zeta(-ni) except that the sign of the imaginary part changes. That means that their absolute values are in fact the same.
- MathPhDLv 61 decade ago
I know that zero is not one of them!
Wolframalpha gives no obvious nice values for first few integers.
All non-trivial zeros of zeta are known to have real part in (0, 1).
Values at 1/2 - u + iv are related to values at 1/2 + u + iv, so your values (which have u=1/2) are related to those at 1+iv. These cannot be calculated by a direct application to the infinite sum, which works only for real part greater than 1, but there are further formulas for getting the values.
