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Fungrim entry: 312147

L ⁣(s,χ)k=1N1χ ⁣(k)ksζ ⁣(Re ⁣(s),N)\left|L\!\left(s, \chi\right) - \sum_{k=1}^{N - 1} \frac{\chi\!\left(k\right)}{{k}^{s}}\right| \le \zeta\!\left(\operatorname{Re}\!\left(s\right), N\right)
Assumptions:qZ1andχGqandsCandRe ⁣(s)>1andNZ1q \in \mathbb{Z}_{\ge 1} \,\mathbin{\operatorname{and}}\, \chi \in G_{q} \,\mathbin{\operatorname{and}}\, s \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \operatorname{Re}\!\left(s\right) > 1 \,\mathbin{\operatorname{and}}\, N \in \mathbb{Z}_{\ge 1}
TeX:
\left|L\!\left(s, \chi\right) - \sum_{k=1}^{N - 1} \frac{\chi\!\left(k\right)}{{k}^{s}}\right| \le \zeta\!\left(\operatorname{Re}\!\left(s\right), N\right)

q \in \mathbb{Z}_{\ge 1} \,\mathbin{\operatorname{and}}\, \chi \in G_{q} \,\mathbin{\operatorname{and}}\, s \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \operatorname{Re}\!\left(s\right) > 1 \,\mathbin{\operatorname{and}}\, N \in \mathbb{Z}_{\ge 1}
Definitions:
Fungrim symbol Notation Short description
Absz\left|z\right| Absolute value
DirichletLL ⁣(s,χ)L\!\left(s, \chi\right) Dirichlet L-function
Sumnf ⁣(n)\sum_{n} f\!\left(n\right) Sum
Powab{a}^{b} Power
HurwitzZetaζ ⁣(s,a)\zeta\!\left(s, a\right) Hurwitz zeta function
ReRe ⁣(z)\operatorname{Re}\!\left(z\right) Real part
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
DirichletGroupGqG_{q} Dirichlet characters with given modulus
CCC\mathbb{C} Complex numbers
Source code for this entry:
Entry(ID("312147"),
    Formula(LessEqual(Abs(Sub(DirichletL(s, chi), Sum(Div(chi(k), Pow(k, s)), Tuple(k, 1, Sub(N, 1))))), HurwitzZeta(Re(s), N))),
    Variables(q, chi, s, N),
    Assumptions(And(Element(q, ZZGreaterEqual(1)), Element(chi, DirichletGroup(q)), Element(s, CC), Greater(Re(s), 1), Element(N, ZZGreaterEqual(1)))))

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Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2019-08-19 14:38:23.809000 UTC