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Results tagged with riemann-zeta-function
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user 536861
The Riemann zeta function is the function of one complex variable $s$ defined by the series $\zeta(s) = \sum_{n \geq 1} \frac{1}{n^s}$ when $\operatorname{Re}(s)>1$. It admits a meromorphic continuation to $\mathbb{C}$ with only a simple pole at $1$. This function satisfies a functional equation relating the values at $s$ and $1-s$. This is the most simple example of an $L$-function and a central object of number theory.
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Show that the function $(n+2)\zeta(n+3)-\zeta(n+2)-n-1$ is positive on $\mathbb{N}.$
I'm trying to prove the positivity of the function $(n+2)\zeta(n+3)-\zeta(n+2)-n-1$ on $\mathbb{N}$ using the inequalities
\begin{equation*}
\frac{s+1}{s}\zeta(s)\zeta(s+2)\geq \zeta^2(s+1),\quad s>1
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Closed form formula
Let $m$ and $n$ are positive integers, then find the sum of the infinite series defined as $$\sum_{k=0}^\infty \frac{(-1)^k\Gamma(m+k) }{\Gamma(m)k!(k+m)^n}.$$ I was managed to the sum with $m=2$ and …
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