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Fungrim entry: 4911bd

1L ⁣(s,χ)p<N(1χ(p)ps)ζ ⁣(Re(s),N)\left|\frac{1}{L\!\left(s, \chi\right)} - \prod_{p < N} \left(1 - \frac{\chi(p)}{{p}^{s}}\right)\right| \le \zeta\!\left(\operatorname{Re}(s), N\right)
Assumptions:qZ1  and  χGq  and  sC  and  Re(s)>1  and  NZ1q \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; \chi \in G_{q} \;\mathbin{\operatorname{and}}\; s \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(s) > 1 \;\mathbin{\operatorname{and}}\; N \in \mathbb{Z}_{\ge 1}
\left|\frac{1}{L\!\left(s, \chi\right)} - \prod_{p < N} \left(1 - \frac{\chi(p)}{{p}^{s}}\right)\right| \le \zeta\!\left(\operatorname{Re}(s), 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}(s) > 1 \;\mathbin{\operatorname{and}}\; N \in \mathbb{Z}_{\ge 1}
Fungrim symbol Notation Short description
Absz\left|z\right| Absolute value
DirichletLL ⁣(s,χ)L\!\left(s, \chi\right) Dirichlet L-function
PrimeProductpf(p)\prod_{p} f(p) Product over primes
Powab{a}^{b} Power
HurwitzZetaζ ⁣(s,a)\zeta\!\left(s, a\right) Hurwitz zeta function
ReRe(z)\operatorname{Re}(z) 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:
    Formula(LessEqual(Abs(Sub(Div(1, DirichletL(s, chi)), PrimeProduct(Parentheses(Sub(1, Div(chi(p), Pow(p, s)))), For(p), Less(p, N)))), 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.

2021-03-15 19:12:00.328586 UTC