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  "body": "\n## Definition\n\nLet $a \\colon \\set{1, 2, 3, \\dots} \\to \\C$ and let $F(s) = \\sum_{n \\ge 1} a(n) n^{-s}$ be its [[Dirichlet series|8MWTEF6462]].\nLet $\\sigma_a$ be its abscissa of [[absolute convergence|0YBDSJARWN]], and assume $\\sigma_a \u003c \\infty$.\nThe *Mellin representations* of $F$ are the two identities below, which hold for every $s \\in \\C$ with $\\operatorname{Re} s \u003e \\max(\\sigma_a, 0)$.\n\n1. With $f_a(x) = \\sum_{n \\ge 1} a(n) e^{-nx}$ for $x \u003e 0$,\n$$\\Gamma(s) F(s) = \\int_0^\\infty x^{s-1} f_a(x) \\, dx = \\mathcal{M}f_a(s).$$\n2. With the summatory function $A(x) = \\sum_{n \\le x} a(n)$ for $x \u003e 0$,\n$$\\frac{F(s)}{s} = \\int_1^\\infty A(x) \\, x^{-s-1} \\, dx = \\mathcal{M}A(-s).$$\n\nHere $\\mathcal{M}$ is the [[Mellin transform|N5PVGRJE1T]] and $\\Gamma$ is the [[gamma function|0H5SYBSED9]].\n\n## Remarks\n\nFor identity 1, $\\int_0^\\infty x^{s-1} e^{-nx} \\, dx = \\Gamma(s) n^{-s}$ when $\\operatorname{Re} s \u003e 0$, and the sum of the absolute values $\\Gamma(\\operatorname{Re} s) \\sum_n \\abs{a(n)} n^{-\\operatorname{Re} s}$ is finite when $\\operatorname{Re} s \u003e \\sigma_a$, so sum and integral may be exchanged.\nFor identity 2, $A(x) = 0$ for $x \u003c 1$, and $\\int_n^\\infty x^{-s-1} \\, dx = n^{-s}/s$; exchanging sum and integral is justified the same way.\nExample: $a(n) = 1$ gives $\\sigma_a = 1$, $f_a(x) = 1/(e^{x} - 1)$ and $A(x) = \\floor{x}$, so\n$$\\Gamma(s) \\zeta(s) = \\int_0^\\infty \\frac{x^{s-1}}{e^{x} - 1} \\, dx, \\qquad \\frac{\\zeta(s)}{s} = \\int_1^\\infty \\floor{x} \\, x^{-s-1} \\, dx, \\qquad \\operatorname{Re} s \u003e 1.$$\nNon-example: at $s = 1$ with $a(n) = 1$, the condition $\\operatorname{Re} s \u003e \\sigma_a$ fails, and indeed $\\int_0^\\infty dx/(e^x - 1)$ diverges at $0$.\n",
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