Could you show that in a right neighbourhood of $x=0$ $$\frac{1}{n!}\frac{d^n}{dx^n}\Gamma(x)\zeta(x)$$
behaves like $(-1)^n \zeta (-1)+1$ when $n\to \infty$?
Could you show that in a right neighbourhood of $x=0$ $$\frac{1}{n!}\frac{d^n}{dx^n}\Gamma(x)\zeta(x)$$
behaves like $(-1)^n \zeta (-1)+1$ when $n\to \infty$?
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