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  "body": "\n## Statement\n\nLet $f$ be a [[multiplicative function|0AP80TT660]] from the positive integers to $\\C$ such that $\\sum_{n=1}^{\\infty} f(n)$ converges [[absolutely|0W3K3QPEWT]].\nThen for each [[prime|0SN0MAEKYX]] $p$ the series $\\sum_{k=0}^{\\infty} f(p^k)$ converges absolutely, and\n$$\\sum_{n=1}^{\\infty} f(n) = \\lim_{x \\to \\infty} \\prod_{p \\le x} \\sum_{k=0}^{\\infty} f(p^k),$$\nwhere the [[product|RZ0Z2BTMT4]] runs over the primes $p \\le x$.\nIf $f$ is completely multiplicative, then $\\abs{f(p)} \u003c 1$ for each prime $p$, and the factor for $p$ equals $1/(1 - f(p))$.\n\n## Remarks\n\nFor finite $x$, multiplying out the finitely many absolutely convergent factors gives $\\sum f(n)$ over all $n$ whose prime factors are at most $x$.\nHere one uses that $f$ is multiplicative and the [[fundamental theorem of arithmetic|084HH1NDCB]].\nEvery missing $n$ is larger than $x$, so the difference from the full sum is at most $\\sum_{n \u003e x} \\abs{f(n)}$, which tends to $0$.\nWith $f(n) = n^{-s}$ and $\\operatorname{Re}(s) \u003e 1$ this is the [[Euler product formula|0C3PN4075R]] for the zeta function.\nWith $f(n) = \\mu(n) n^{-s}$, where $\\mu$ is the [[Möbius function|0C8QENHYZW]], it gives $\\sum \\mu(n) n^{-s} = \\prod_p (1 - p^{-s})$.\n\n## Explanations\n\n### L1 human\n\nAn Euler product writes a sum over all positive integers as a product over the primes.\nIt works because each integer is built from primes in exactly one way, and a multiplicative function respects this building.\nIt is the basic bridge between sums over integers and the primes.\n\n### L2 human\n\nEuler products are the starting point of analytic number theory.\nThe zeta function, Dirichlet $L$-functions and many other Dirichlet series have them in their half-plane of absolute convergence.\nFrom the product one reads off facts about primes and zeros.\nFor example, the zeta function has no zeros where its Euler product converges, since a convergent product of nonzero factors is nonzero.\n",
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  "name": "Euler product for multiplicative functions",
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