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A zeta limit
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Author:  Tolaso J Kos [ Tue Apr 11, 2017 7:30 pm ]
Post subject:  A zeta limit

Let us denote with $\zeta$ the Riemann zeta function defined as $\zeta(0)=-\frac{1}{2}$. Let us also denote with $\zeta^{(n)}$ the $n$-th derivative of zeta. Evaluate the limit

$$\ell=\lim_{n \rightarrow +\infty} \frac{\zeta^{(n)}(0)}{n!}$$

Author:  Riemann [ Sat May 26, 2018 1:28 pm ]
Post subject:  Re: A zeta limit

We know that the function $\displaystyle f(z) \equiv \zeta(z) + \frac{1}{1-z}$ is a holomorphic function. The Taylor series around $0$ is

$$\zeta(z) + \frac{1}{1-z} = \sum_{n=0}^{\infty} \left( \frac{\zeta^{(n)}(0)}{n!} + 1 \right) z^n$$

which converges forall $z \in \mathbb{C}$ thus $\displaystyle \lim_{n \to +\infty} \frac{\zeta^{(n)}(0)}{n!} = -1$.

Author:  Riemann [ Sat May 26, 2018 1:31 pm ]
Post subject:  Re: A zeta limit

Using the above fact we get that $\zeta^{(n)}(0) \sim -n!$.

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