Fungrim entry: f88596

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

TeX:

\lim_{N \to \infty} \frac{\sum_{q=1}^{N} \# G_{q}}{\frac{1}{2} {N}^{2}} = \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
`DirichletGroup`	$G_{q}$	Dirichlet characters with given modulus
`Pow`	${a}^{b}$	Power
`Infinity`	$\infty$	Positive infinity
`Pi`	$\pi$	The constant pi (3.14...)

Source code for this entry:

Entry(ID("f88596"),
    Formula(Equal(SequenceLimit(Div(Sum(Cardinality(DirichletGroup(q)), For(q, 1, N)), Mul(Div(1, 2), Pow(N, 2))), For(N, Infinity)), Div(6, Pow(Pi, 2)))))

Topics using this entry

Dirichlet characters

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