# Fungrim entry: 3b43b0

$\lim_{N \to \infty} \frac{\sum_{q=1}^{N} \# G_{q}^{\text{primitive}}}{\sum_{q=1}^{N} \# G_{q}} = \frac{6}{{\pi}^{2}}$
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
SequenceLimit$\lim_{n \to a} f(n)$ Limiting value of sequence
Sum$\sum_{n} f(n)$ Sum
Cardinality$\# S$ Set cardinality
PrimitiveDirichletCharacters$G_{q}^{\text{primitive}}$ 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

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

2019-12-30 15:00:46.909060 UTC