# 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

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