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Fungrim entry: 03fbe8

χq(,n)=exp ⁣(2πi((1x)(1y)8+ab2e2))   where q=2e,L ⁣(k)={(1,log5 ⁣(k)modq),k{5imodq:iZ1}(1,log5 ⁣(k)modq),k{5imodq:iZ1},(x,a)=L ⁣(),(y,b)=L ⁣(n)\chi_{q}(\ell, n) = \exp\!\left(2 \pi i \left(\frac{\left(1 - x\right) \left(1 - y\right)}{8} + \frac{a b}{{2}^{e - 2}}\right)\right)\; \text{ where } q = {2}^{e},\,L\!\left(k\right) = \begin{cases} \left(1, \log_{5}\!\left(k\right) \bmod q\right), & k \in \left\{ {5}^{i} \bmod q : i \in \mathbb{Z}_{\ge 1} \right\}\\\left(-1, \log_{5}\!\left(-k\right) \bmod q\right), & k \in \left\{ -{5}^{i} \bmod q : i \in \mathbb{Z}_{\ge 1} \right\}\\ \end{cases},\,\left(x, a\right) = L\!\left(\ell\right),\,\left(y, b\right) = L\!\left(n\right)
Assumptions:eZ3and{1,2,2e1}andnZandgcd ⁣(,2)=gcd ⁣(n,2)=1e \in \mathbb{Z}_{\ge 3} \,\mathbin{\operatorname{and}}\, \ell \in \{1, 2, \ldots {2}^{e} - 1\} \,\mathbin{\operatorname{and}}\, n \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, \gcd\!\left(\ell, 2\right) = \gcd\!\left(n, 2\right) = 1
TeX:
\chi_{q}(\ell, n) = \exp\!\left(2 \pi i \left(\frac{\left(1 - x\right) \left(1 - y\right)}{8} + \frac{a b}{{2}^{e - 2}}\right)\right)\; \text{ where } q = {2}^{e},\,L\!\left(k\right) = \begin{cases} \left(1, \log_{5}\!\left(k\right) \bmod q\right), & k \in \left\{ {5}^{i} \bmod q : i \in \mathbb{Z}_{\ge 1} \right\}\\\left(-1, \log_{5}\!\left(-k\right) \bmod q\right), & k \in \left\{ -{5}^{i} \bmod q : i \in \mathbb{Z}_{\ge 1} \right\}\\ \end{cases},\,\left(x, a\right) = L\!\left(\ell\right),\,\left(y, b\right) = L\!\left(n\right)

e \in \mathbb{Z}_{\ge 3} \,\mathbin{\operatorname{and}}\, \ell \in \{1, 2, \ldots {2}^{e} - 1\} \,\mathbin{\operatorname{and}}\, n \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, \gcd\!\left(\ell, 2\right) = \gcd\!\left(n, 2\right) = 1
Definitions:
Fungrim symbol Notation Short description
DirichletCharacterχq(,)\chi_{q}(\ell, \cdot) Dirichlet character
Expez{e}^{z} Exponential function
ConstPiπ\pi The constant pi (3.14...)
ConstIii Imaginary unit
Powab{a}^{b} Power
DiscreteLoglogb ⁣(x)modq\log_{b}\!\left(x\right) \bmod q Discrete logarithm
SetBuilder{f ⁣(x):P ⁣(x)}\left\{ f\!\left(x\right) : P\!\left(x\right) \right\} Set comprehension
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
ZZBetween{a,a+1,b}\{a, a + 1, \ldots b\} Integers between a and b inclusive
ZZZ\mathbb{Z} Integers
GCDgcd ⁣(a,b)\gcd\!\left(a, b\right) Greatest common divisor
Source code for this entry:
Entry(ID("03fbe8"),
    Formula(Where(Equal(DirichletCharacter(q, ell, n), Exp(Mul(Mul(Mul(2, ConstPi), ConstI), Add(Div(Mul(Sub(1, x), Sub(1, y)), 8), Div(Mul(a, b), Pow(2, Sub(e, 2))))))), Equal(q, Pow(2, e)), Equal(L(k), Cases(Tuple(Tuple(1, DiscreteLog(k, 5, q)), Element(k, SetBuilder(Mod(Pow(5, i), q), i, Element(i, ZZGreaterEqual(1))))), Tuple(Tuple(-1, DiscreteLog(Neg(k), 5, q)), Element(k, SetBuilder(Mod(Neg(Pow(5, i)), q), i, Element(i, ZZGreaterEqual(1))))))), Equal(Tuple(x, a), L(ell)), Equal(Tuple(y, b), L(n)))),
    Variables(e, ell, n),
    Assumptions(And(Element(e, ZZGreaterEqual(3)), Element(ell, ZZBetween(1, Sub(Pow(2, e), 1))), Element(n, ZZ), Equal(GCD(ell, 2), GCD(n, 2), 1))))

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

2019-08-17 11:32:46.829430 UTC