Fungrim entry: 291569

\frac{1}{L\!\left(s, \chi\right)} = \sum_{n=1}^{\infty} \frac{\mu(n) \chi(n)}{{n}^{s}}

Assumptions:

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

TeX:

\frac{1}{L\!\left(s, \chi\right)} = \sum_{n=1}^{\infty} \frac{\mu(n) \chi(n)}{{n}^{s}}

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

Definitions:

Fungrim symbol	Notation	Short description
`DirichletL`	$L\!\left(s, \chi\right)$	Dirichlet L-function
`Sum`	$\sum_{n} f(n)$	Sum
`MoebiusMu`	$\mu(n)$	Möbius function
`Pow`	${a}^{b}$	Power
`Infinity`	$\infty$	Positive infinity
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`DirichletGroup`	$G_{q}$	Dirichlet characters with given modulus
`CC`	$\mathbb{C}$	Complex numbers
`Re`	$\operatorname{Re}(z)$	Real part

Source code for this entry:

Entry(ID("291569"),
    Formula(Equal(Div(1, DirichletL(s, chi)), Sum(Div(Mul(MoebiusMu(n), chi(n)), Pow(n, s)), For(n, 1, Infinity)))),
    Variables(q, chi, s),
    Assumptions(And(Element(q, ZZGreaterEqual(1)), Element(chi, DirichletGroup(q)), Element(s, CC), Greater(Re(s), 1))))

Topics using this entry

Dirichlet L-functions