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  "body": "\n## Definition\n\nLet $(\\lambda_n)_{n \\ge 0}$ be a strictly increasing sequence of real numbers with $\\lambda_0 \\ge 0$ and $\\lambda_n \\to \\infty$, and let $\\delta \u003e 0$.\nLet $\\sum_{n \\ge 0} a_n$ be a [[series|0JH4MN4K1C]] of complex numbers.\nFor real $\\omega \u003e 0$, the *Riesz mean* of type $\\lambda$ and order $\\delta$ is the finite sum\n$$R^{\\delta}_{\\lambda}(\\omega) = \\sum_{n \\,:\\, \\lambda_n \u003c \\omega} \\left(1 - \\frac{\\lambda_n}{\\omega}\\right)^{\\delta} a_n.$$\nThe series is $(R, \\lambda, \\delta)$ summable to $s$ if $R^{\\delta}_{\\lambda}(\\omega) \\to s$ as $\\omega \\to \\infty$.\n\n## Remarks\n\nExample: take $\\lambda_n = n$, $\\delta = 1$ and an integer cut-off $\\omega = N + 1$.\nThen\n$$R^1_{\\lambda}(N + 1) = \\sum_{n=0}^{N} \\frac{N + 1 - n}{N + 1} a_n = \\frac{s_0 + s_1 + \\cdots + s_N}{N + 1},$$\nwhere $s_j$ are the [[partial sums|0NB643K60Q]], because $a_n$ occurs in exactly the $N + 1 - n$ partial sums $s_n, \\dots, s_N$.\nSo these Riesz means are the Cesàro means of the partial sums.\n\nNon-example: if $\\lambda_n = 1 - 2^{-n}$, the sequence is increasing but bounded, so the condition $\\lambda_n \\to \\infty$ fails.\nThen for $\\omega \u003e 1$ every term enters $R^{\\delta}_{\\lambda}(\\omega)$, which is not a finite sum, and the definition does not apply.\n\nFor Dirichlet series $\\sum a_n n^{-s}$ one usually takes $\\lambda_n = \\ln n$ for $n \\ge 1$.\n",
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