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  "body": "\n## Definition\n\nLet $\\lambda \\in \\R$, let $\\alpha$ be a real number with $0 \u003c \\alpha \\le 1$, and let $s \\in \\C$.\nThe *Lerch zeta function* is the generalized [[Dirichlet series|8MWTEF6462]]\n$$L(\\lambda, \\alpha, s) = \\sum_{n=0}^{\\infty} \\frac{e^{2\\pi i \\lambda n}}{(n + \\alpha)^s}.$$\nIt converges for $\\operatorname{Re} s \u003e 1$, and also for $\\operatorname{Re} s \u003e 0$ when $\\lambda \\notin \\Z$.\n\n## Remarks\n\nThe *Lerch transcendent* is $\\Phi(z, s, a) = \\sum_{n \\ge 0} z^n (n + a)^{-s}$, defined for $\\abs{z} \u003c 1$ and $a \\notin \\set{0, -1, -2, \\ldots}$, and for $\\abs{z} = 1$ when $\\operatorname{Re} s \u003e 1$.\nThe two are related by $L(\\lambda, \\alpha, s) = \\Phi(e^{2\\pi i \\lambda}, s, \\alpha)$.\n\nExample: for $\\lambda \\in \\Z$ every phase factor is $1$, so $L(\\lambda, \\alpha, s) = \\zeta(s, \\alpha)$ is the [[Hurwitz zeta function|G4SW7HJG88]], and $L(0, 1, s) = \\zeta(s)$.\nWith $a = 1$ the transcendent gives the [[polylogarithm|96CHDZKGQ0]]: $z \\, \\Phi(z, s, 1) = \\sum_{n \\ge 0} z^{n+1} / (n+1)^s = \\operatorname{Li}_s(z)$.\n\nNon-example: $\\lambda = 0$, $\\alpha = 1$, $s = 1$ gives the harmonic series $\\sum 1/(n+1)$, which diverges; the condition $\\operatorname{Re} s \u003e 1$ is needed when $\\lambda \\in \\Z$.\nFor $\\lambda = \\frac12$ and the same $\\alpha, s$, the series is $1 - \\frac12 + \\frac13 - \\cdots = \\log 2$, which converges.\n\nAs a function of $s$, $L(\\lambda, \\alpha, s)$ extends to a meromorphic function on $\\C$; for $\\lambda \\notin \\Z$ it is entire.\n",
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