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  "body": "\n## Summary\n\nZeta function regularization assigns finite values to divergent sums $\\sum \\lambda_n$ and divergent products $\\prod \\lambda_n$ of positive numbers.\nOne forms the zeta function $\\zeta_\\lambda(s) = \\sum \\lambda_n^{-s}$, which converges for large $\\operatorname{Re} s$, and continues it by [[analytic continuation|0M3YQJ81NB]].\nThe regularized sum is $\\zeta_\\lambda(-1)$, and the regularized product is $e^{-\\zeta_\\lambda'(0)}$.\nFor $\\lambda_n = n$ the [[Riemann zeta function|1XWBY4N0VT]] gives the regularized value $-\\frac{1}{12}$ of $1 + 2 + 3 + \\cdots$.\nThe method is used for determinants of Laplacians and in quantum field theory.\n\n## Explanations\n\n### L1 human\n\nThe sum $1 + 2 + 3 + \\cdots$ diverges, but the sum $1^{-s} + 2^{-s} + 3^{-s} + \\cdots$ converges when $s \u003e 1$ and defines a function of $s$.\nThat function has a unique analytic continuation to all complex $s \\ne 1$.\nPlugging $s = -1$ into the extension formally gives back $1 + 2 + 3 + \\cdots$, and the value obtained is $-\\frac{1}{12}$.\nZeta regularization does the same for any sequence of positive numbers that grows fast enough.\n",
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