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Question Number 130508 by Raxreedoroid last updated on 26/Jan/21

$$lim{}_{n\to \mathrm{\infty }}\underset{k=1}{\sum ^n}\frac{cos(x\cdot ln\left(k\right))}{\sqrt{k}}=?$$$$wherex\in \mathbb{R}$$

Answered by Dwaipayan Shikari last updated on 26/Jan/21

$$\underset{k=1}{\sum ^{\mathrm{\infty }}}\frac{cos\left(log\left(k^x\right)\right)}{\sqrt{k}}=\frac{1}{2}\underset{k=1}{\sum ^{\mathrm{\infty }}}\frac{k^{xi}+k^{-xi}}{\sqrt{k}}=\frac{1}{2}\zeta (\frac{1}{2}-xi)+\frac{1}{2}\zeta (\frac{1}{2}+xi)=0$$$$As\zeta (\frac{1}{2}+\mathrm{\Phi }i)=0(\mathrm{\Phi }\in \mathbb{R})$$

Commented by Raxreedoroid last updated on 26/Jan/21

$${Isn}^{\prime }tRiemannhypothisis{hasn}^{\prime }beenprovenyet?$$

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