Fungrim entry: d69b41

\sum_{a=1}^{q} \chi(a) \frac{z {e}^{a z}}{{e}^{q z} - 1} = \sum_{n=0}^{\infty} B_{n,\chi} \frac{{z}^{n}}{n !}

Assumptions:

q \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; \chi \in G_{q} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \ne 0 \;\mathbin{\operatorname{and}}\; \left|z\right| < \frac{2 \pi}{q}

TeX:

\sum_{a=1}^{q} \chi(a) \frac{z {e}^{a z}}{{e}^{q z} - 1} = \sum_{n=0}^{\infty} B_{n,\chi} \frac{{z}^{n}}{n !}

q \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; \chi \in G_{q} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \ne 0 \;\mathbin{\operatorname{and}}\; \left|z\right| < \frac{2 \pi}{q}

Definitions:

Fungrim symbol	Notation	Short description
`Sum`	$\sum_{n} f(n)$	Sum
`Exp`	${e}^{z}$	Exponential function
`GeneralizedBernoulliB`	$B_{n,\chi}$	Generalized Bernoulli number
`Pow`	${a}^{b}$	Power
`Factorial`	$n !$	Factorial
`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
`Abs`	$\left\|z\right\|$	Absolute value
`Pi`	$\pi$	The constant pi (3.14...)

Source code for this entry:

Entry(ID("d69b41"),
    Formula(Equal(Sum(Mul(chi(a), Div(Mul(z, Exp(Mul(a, z))), Sub(Exp(Mul(q, z)), 1))), For(a, 1, q)), Sum(Mul(GeneralizedBernoulliB(n, chi), Div(Pow(z, n), Factorial(n))), For(n, 0, Infinity)))),
    Variables(q, chi, z),
    Assumptions(And(Element(q, ZZGreaterEqual(1)), Element(chi, DirichletGroup(q)), Element(z, CC), NotEqual(z, 0), Less(Abs(z), Div(Mul(2, Pi), q)))))

Topics using this entry

Dirichlet L-functions