Fungrim entry: 4cf4e4

\chi_{q \, . \, \ell}\!\left(n\right) = \exp\!\left(\frac{2 \pi i a b}{\varphi(q)}\right)\; \text{ where } q = {p}^{e},\;g = g_{p},\;a = \log_{g}\!\left(\ell\right) \bmod q,\;b = \log_{g}\!\left(n\right) \bmod q

Assumptions:

p \in \mathbb{P} \;\mathbin{\operatorname{and}}\; p \ge 3 \;\mathbin{\operatorname{and}}\; e \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; \ell \in \{1, 2, \ldots, {p}^{e} - 1\} \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; \gcd\!\left(\ell, {p}^{e}\right) = \gcd\!\left(n, {p}^{e}\right) = 1

TeX:

\chi_{q \, . \, \ell}\!\left(n\right) = \exp\!\left(\frac{2 \pi i a b}{\varphi(q)}\right)\; \text{ where } q = {p}^{e},\;g = g_{p},\;a = \log_{g}\!\left(\ell\right) \bmod q,\;b = \log_{g}\!\left(n\right) \bmod q

p \in \mathbb{P} \;\mathbin{\operatorname{and}}\; p \ge 3 \;\mathbin{\operatorname{and}}\; e \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; \ell \in \{1, 2, \ldots, {p}^{e} - 1\} \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; \gcd\!\left(\ell, {p}^{e}\right) = \gcd\!\left(n, {p}^{e}\right) = 1

Definitions:

Fungrim symbol	Notation	Short description
`DirichletCharacter`	$\chi_{q \, . \, \ell}$	Dirichlet character
`Exp`	${e}^{z}$	Exponential function
`Pi`	$\pi$	The constant pi (3.14...)
`ConstI`	$i$	Imaginary unit
`Totient`	$\varphi(n)$	Euler totient function
`Pow`	${a}^{b}$	Power
`ConreyGenerator`	$g_{p}$	Conrey generator
`DiscreteLog`	$\log_{b}\!\left(x\right) \bmod q$	Discrete logarithm
`PP`	$\mathbb{P}$	Prime numbers
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`Range`	$\{a, a + 1, \ldots, b\}$	Integers between given endpoints
`ZZ`	$\mathbb{Z}$	Integers
`GCD`	$\gcd\!\left(a, b\right)$	Greatest common divisor

Source code for this entry:

Entry(ID("4cf4e4"),
    Formula(Where(Equal(DirichletCharacter(q, ell, n), Exp(Div(Mul(Mul(Mul(Mul(2, Pi), ConstI), a), b), Totient(q)))), Equal(q, Pow(p, e)), Equal(g, ConreyGenerator(p)), Equal(a, DiscreteLog(ell, g, q)), Equal(b, DiscreteLog(n, g, q)))),
    Variables(p, e, ell, n),
    Assumptions(And(Element(p, PP), GreaterEqual(p, 3), Element(e, ZZGreaterEqual(1)), Element(ell, Range(1, Sub(Pow(p, e), 1))), Element(n, ZZ), Equal(GCD(ell, Pow(p, e)), GCD(n, Pow(p, e)), 1))))

Topics using this entry

Dirichlet characters