mathimatikoi.orghttps://www.mathimatikoi.org/forum/ A zeta limithttps://www.mathimatikoi.org/forum/viewtopic.php?f=4&t=1147 Page 1 of 1

 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 $\zeta^{(n)}(0) \sim -n!$.

 Page 1 of 1 All times are UTC [ DST ] Powered by phpBB® Forum Software © phpBB Grouphttps://www.phpbb.com/