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Tagged with binomial-coefficients riemann-zeta-function
5 questions
2
votes
0
answers
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Question on globally convergent formulas for the Riemann zeta function $\zeta(s)$
Consider the following two formulas for $\zeta(s)$
$$\zeta(s)=\underset{K\to\infty}{\text{lim}}\left(\frac{1}{1-2^{1-s}}\sum\limits_{n=0}^K \frac{1}{2^{n+1}}\sum\limits_{k=0}^n \binom{n}{k} \frac{(-1)^...
- 1,126
10
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1
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What is known about sums of the form $\sum_{n=2}^{\infty}[\zeta(n)-1]^{p} $?
A fair bit is known about rational zeta series. This includes identities like $$ \sum_{n=2}^{\infty} [\zeta(n) -1] = 1 . $$
Many more identities can be found in articles by e.g. Borwein and Adamchik &...
- 4,829
1
vote
1
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Are there variations of Ramaswami's formula for the analytic continuation of the Riemann zeta function?
On p. 286 of Borwein's paper entitled "Computational Strategies for the Riemann zeta function", the author mentions a formula due to Ramaswami: $$(1-2^{1-s})\zeta(s) = \sum_{n=1}^{\infty} \...
- 4,829
3
votes
3
answers
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Show that the ratio of limits converges to the nearest Riemann zeta zero except when the ratio is a singularity
Let $h(s,n)$ be:
$$h(s,n)=\lim_{c\to 1} \, \frac{(-1)^{n-2}}{(n-2)!}\zeta (c)^{n-2} \sum _{k=1}^{n-1} \frac{(-1)^{k-1} \binom{n-2}{k-1}}{\zeta ((c-1) (k-1)+s)}$$
and let $g(s,n)$ be:
$$g(s,n)=\lim_{c\...
- 1,183
2
votes
1
answer
194
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Does this series, related to the Hasse/Ser series for $\zeta(s)$, converge for all $s \in \mathbb{C}$?
I have asked this question at math stack exchange, however it did not get any traction. Still curious about the answer though.
Numerical evidence suggests that:
$$\lim_{N \to +\infty} \sum_{n=1}^N\...
- 128k