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  "body": "\n## Definition\n\nLet $a$ be a real number with $a \u003e 0$, and let $s$ be a complex number with $\\operatorname{Re} s \u003e 1$.\nThe *Hurwitz zeta function* is\n$$\\zeta(s, a) = \\sum_{n=0}^{\\infty} \\frac{1}{(n + a)^{s}},$$\nwhere $(n + a)^{s} = e^{s \\log(n + a)}$ with the real logarithm.\nThe series converges absolutely for $\\operatorname{Re} s \u003e 1$.\nFor fixed $a$, the function $s \\mapsto \\zeta(s, a)$ has an [[analytic continuation|0M3YQJ81NB]] to a [[meromorphic function|0ZZ7ZYDR8W]] on $\\C$, and this continuation is also written $\\zeta(s, a)$.\n\n## Remarks\n\nFor $a = 1$, the series is $\\sum_{n \\ge 1} n^{-s}$, so $\\zeta(s, 1)$ is the [[Riemann zeta function|1XWBY4N0VT]] $\\zeta(s)$.\nFor $a = 1/2$, the terms are $2^s (2n + 1)^{-s}$, so $\\zeta(s, 1/2) = 2^s \\sum_{m \\text{ odd}} m^{-s} = (2^s - 1)\\zeta(s)$.\nThe value $s = 1$ is not allowed in the series: there the terms behave like $1/n$ and the series diverges.\nThe continuation has a single pole, at $s = 1$; it is simple with residue $1$.\nFor a [[Dirichlet character|7XB5VJKDPG]] $\\chi$ modulo $q$, grouping $n$ by its residue gives\n$$L(s, \\chi) = q^{-s} \\sum_{r=1}^{q} \\chi(r)\\, \\zeta\\!\\left(s, \\frac{r}{q}\\right).$$\nSome authors also allow complex $a$ with $\\operatorname{Re} a \u003e 0$.\n",
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