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Fungrim entry: 3b43b0

limNq=1N#Gqprimitiveq=1N#Gq=6π2\lim_{N \to \infty} \frac{\sum_{q=1}^{N} \# G_{q}^{\text{primitive}}}{\sum_{q=1}^{N} \# G_{q}} = \frac{6}{{\pi}^{2}}
References:
  • H. Jager, On the number of Dirichlet characters with modulus not exceeding x, Indagationes Mathematicae, Volume 76, Issue 5, 1973, Pages 452-455, https://doi.org/10.1016/1385-7258(73)90069-3
TeX:
\lim_{N \to \infty} \frac{\sum_{q=1}^{N} \# G_{q}^{\text{primitive}}}{\sum_{q=1}^{N} \# G_{q}} = \frac{6}{{\pi}^{2}}
Definitions:
Fungrim symbol Notation Short description
SequenceLimitlimnaf(n)\lim_{n \to a} f(n) Limiting value of sequence
Sumnf(n)\sum_{n} f(n) Sum
Cardinality#S\# S Set cardinality
PrimitiveDirichletCharactersGqprimitiveG_{q}^{\text{primitive}} Primitive Dirichlet characters with given modulus
DirichletGroupGqG_{q} Dirichlet characters with given modulus
Infinity\infty Positive infinity
Powab{a}^{b} Power
Piπ\pi The constant pi (3.14...)
Source code for this entry:
Entry(ID("3b43b0"),
    Formula(Equal(SequenceLimit(Div(Sum(Cardinality(PrimitiveDirichletCharacters(q)), For(q, 1, N)), Sum(Cardinality(DirichletGroup(q)), For(q, 1, N))), For(N, Infinity)), Div(6, Pow(Pi, 2)))),
    References("H. Jager, On the number of Dirichlet characters with modulus not exceeding x, Indagationes Mathematicae, Volume 76, Issue 5, 1973, Pages 452-455, https://doi.org/10.1016/1385-7258(73)90069-3"))

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2019-12-30 15:00:46.909060 UTC