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Fungrim entry: 4cf4e4

χq. ⁣(n)=exp ⁣(2πiabφ(q))   where q=pe,g=gp,a=logg ⁣()modq,b=logg ⁣(n)modq\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:pPandp3andeZ1and{1,2,,pe1}andnZandgcd ⁣(,pe)=gcd ⁣(n,pe)=1p \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
DirichletCharacterχq.\chi_{q \, . \, \ell} Dirichlet character
Expez{e}^{z} Exponential function
ConstPiπ\pi The constant pi (3.14...)
ConstIii Imaginary unit
Totientφ(n)\varphi(n) Euler totient function
Powab{a}^{b} Power
ConreyGeneratorgpg_{p} Conrey generator
DiscreteLoglogb ⁣(x)modq\log_{b}\!\left(x\right) \bmod q Discrete logarithm
PPP\mathbb{P} Prime numbers
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
Range{a,a+1,,b}\{a, a + 1, \ldots, b\} Integers between given endpoints
ZZZ\mathbb{Z} Integers
GCDgcd ⁣(a,b)\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, ConstPi), 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))))

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Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2019-10-05 13:11:19.856591 UTC