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  "body": "\n## Definition\n\nLet $a \\colon \\set{1, 2, 3, \\dots} \\to \\C$ be a function, called an arithmetic function.\nFor $s \\in \\C$ and an integer $n \\ge 1$, let $n^{-s} = e^{-s \\log n}$, using the [[complex exponential|07GJV36896]].\nThe *Dirichlet series* of $a$ is the [[series|0NM626FH6Q]]\n$$F(s) = \\sum_{n=1}^{\\infty} \\frac{a(n)}{n^s},$$\nconsidered as a function of $s$ at the points where it converges.\n\n## Remarks\n\nLet $\\sigma_a$ be the infimum of the real $\\sigma$ such that $\\sum_n \\abs{a(n)} n^{-\\sigma}$ converges, the abscissa of [[absolute convergence|0YBDSJARWN]].\nThe series converges absolutely for $\\operatorname{Re} s \u003e \\sigma_a$ and defines a holomorphic function there.\nIt also has an abscissa of convergence $\\sigma_c$, with $\\sigma_c \\le \\sigma_a \\le \\sigma_c + 1$.\nExample: for $a(n) = (-1)^{n+1}$, the series $\\sum (-1)^{n+1} n^{-s}$ converges for $\\operatorname{Re} s \u003e 0$ and converges absolutely for $\\operatorname{Re} s \u003e 1$, so $\\sigma_c = 0$ and $\\sigma_a = 1$.\nNon-example: the power series $\\sum_{n} a_n z^n$ is not a Dirichlet series, since its terms are powers $z^n$ of the variable rather than $n^{-s}$.\nIf $F$ and $G$ are the Dirichlet series of $a$ and $b$, both absolutely convergent at $s$, then $F(s) G(s)$ is the Dirichlet series of the [[Dirichlet convolution|037F589D1D]] $a * b$ at $s$.\nFor example $\\sum_n \\mu(n) n^{-s} = 1/\\zeta(s)$ for $\\operatorname{Re} s \u003e 1$, with $\\mu$ the [[Möbius function|0FANE7P0DY]].\n",
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