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Question Number 142875 by Snail last updated on 06/Jun/21

$$Provethat\zeta \left(s\right)=\prod _{prime}\frac{1}{1-p^{-s}}$$

Answered by Dwaipayan Shikari last updated on 06/Jun/21

$$\zeta \left(s\right)=1+\frac{1}{2^s}+\frac{1}{3^s}+\frac{1}{4^s}+...$$$$\frac{\zeta \left(s\right)}{2^s}=\frac{1}{2^s}+\frac{1}{4^s}+\frac{1}{6^s}+...\left(Multiplesof2\right)$$$$\zeta \left(s\right)(1-\frac{1}{2^s})=1+\frac{1}{3^s}+\frac{1}{5^s}+\frac{1}{7^s}+...\left(Subtract\right)$$$$\zeta \left(s\right)(1-\frac{1}{2^s})\frac{1}{3^s}=\frac{1}{3^s}+\frac{1}{9^s}+\frac{1}{{15}^s}+\frac{1}{{21}^s}+...\left(allmultiplesof3\right)$$$$\zeta \left(s\right)(1-\frac{1}{2^s})-\zeta \left(s\right)(1-\frac{1}{2^s})\frac{1}{3^s}=\frac{1}{1}+\frac{1}{5^s}+\frac{1}{7^s}+..$$$$\zeta \left(s\right)(1-\frac{1}{2^s})(1-\frac{1}{3^s})=1+\frac{1}{5^s}+\frac{1}{7^s}+..$$$$Sinceeachnumbers(\ne 1)arebulidfromprimes,$$$$wecankeepgoinglikethis$$$$$$$$\zeta \left(s\right)(1-\frac{1}{2^s})(1-\frac{1}{3^s})(1-\frac{1}{5^s})..=1$$$$\zeta \left(s\right)=\underset{p}{\prod ^{\mathrm{\infty }}}\frac{1}{1-\frac{1}{p^s}}$$

Commented by Snail last updated on 06/Jun/21

$$Thanks$$

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