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  "body": "\n## Definition\n\nLet $(\\lambda_n)_{n \\ge 1}$ be a [[sequence|0QE8STGJH2]] of real numbers such that:\n\n1. $0 \u003c \\lambda_1 \\le \\lambda_2 \\le \\cdots$ and $\\lambda_n \\to \\infty$;\n2. the series $\\zeta_\\lambda(s) = \\sum_{n \\ge 1} \\lambda_n^{-s}$ converges for all complex $s$ with $\\operatorname{Re} s \u003e \\sigma_0$, for some real $\\sigma_0$;\n3. $\\zeta_\\lambda$ has a meromorphic [[analytic continuation|0M3YQJ81NB]] to a domain that contains $s = -1$ and $s = 0$.\n\nIf the continuation is holomorphic at $s = -1$, the *zeta-regularized sum* of $\\sum \\lambda_n$ is $\\zeta_\\lambda(-1)$.\nIf it is holomorphic at $s = 0$, the *zeta-regularized product* of $\\prod \\lambda_n$ is $\\exp(-\\zeta_\\lambda'(0))$.\n\n## Remarks\n\nHere $\\lambda_n^{-s} = e^{-s \\ln \\lambda_n}$, which needs $\\lambda_n \u003e 0$.\nThe function $\\zeta_\\lambda$ is a generalized [[Dirichlet series|8MWTEF6462]].\n\nExample: for $\\lambda_n = 2^n$ one has $\\zeta_\\lambda(s) = \\sum_{n \\ge 1} 2^{-ns} = \\frac{1}{2^s - 1}$ for $\\operatorname{Re} s \u003e 0$, and the right side is meromorphic on $\\C$.\nSo the regularized sum of $2 + 4 + 8 + \\cdots$ is $\\frac{1}{2^{-1} - 1} = -2$.\nFor $\\lambda_n = n$, $\\zeta_\\lambda$ is the Riemann zeta function, so the regularized sum of $1 + 2 + 3 + \\cdots$ is $\\zeta(-1) = -\\frac{1}{12}$, and the regularized product $\\prod n$ is $e^{-\\zeta'(0)} = \\sqrt{2\\pi}$.\n\nNon-example: the sequence $\\lambda_n = (-1)^n n$ fails clause 1, since it has negative terms and $\\lambda_n^{-s}$ is not defined by a real logarithm.\n",
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