Fungrim entry: 3b43b0

\lim_{N \to \infty} \frac{\sum_{q=1}^{N} \# G^{\text{Primitive}}_{q}}{\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^{\text{Primitive}}_{q}}{\sum_{q=1}^{N} \# G_{q}} = \frac{6}{{\pi}^{2}}

Definitions:

Fungrim symbol	Notation	Short description
`SequenceLimit`	$\lim_{n \to a} f(n)$	Limiting value of sequence
`Sum`	$\sum_{n} f(n)$	Sum
`Cardinality`	$\# S$	Set cardinality
`PrimitiveDirichletCharacters`	$G^{\text{Primitive}}_{q}$	Primitive Dirichlet characters with given modulus
`DirichletGroup`	$G_{q}$	Dirichlet characters with given modulus
`Infinity`	$\infty$	Positive infinity
`Pow`	${a}^{b}$	Power
`Pi`	$\pi$	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"))

Topics using this entry

Dirichlet characters