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  "body": "\n## Statement\n\nLet $s$ be a complex number with real part $\\operatorname{Re}(s) \u003e 1$, and let $\\zeta(s) = \\sum_{n=1}^{\\infty} n^{-s}$ be the Riemann zeta function.\nThen\n$$\\zeta(s) = \\prod_{p} \\frac{1}{1 - p^{-s}},$$\nwhere the product runs over all [[prime numbers|0SN0MAEKYX]] $p$ and converges absolutely.\n\n## Remarks\n\nThe proof expands each factor as a geometric series $1 + p^{-s} + p^{-2s} + \\cdots$. The [[fundamental theorem of arithmetic|084HH1NDCB]] shows that the product of these series contains each term $n^{-s}$ exactly one time.\nThe same method gives a product formula $\\sum_n f(n) n^{-s} = \\prod_p \\sum_{k \\ge 0} f(p^k) p^{-ks}$ for each [[multiplicative function|0AP80TT660]] $f$ when the series converges absolutely.\nSince $\\zeta(s) \\to \\infty$ as $s \\to 1^+$, the formula gives a proof of [[Euclid's theorem|0292EPPPT0]].\n\n## Explanations\n\n### L1 human\n\nMultiply $(1 + \\tfrac12 + \\tfrac14 + \\cdots)(1 + \\tfrac13 + \\tfrac19 + \\cdots)(1 + \\tfrac15 + \\cdots) \\cdots$ and expand.\nEach fraction $1/n$ appears one time, because $n$ is a product of prime powers in exactly one way.\nThe formula with the exponent $s$ is the same idea with $1 / n^s$.\n",
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