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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}\!\left(n\right) = \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(k) = \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(\ell),\;\left(y, b\right) = L(n)
Assumptions:eZ3  and  {1,2,,2e1}  and  nZ  and  gcd ⁣(,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}\!\left(n\right) = \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(k) = \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(\ell),\;\left(y, b\right) = L(n)

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} Dirichlet character
Expez{e}^{z} Exponential function
Piπ\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
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("03fbe8"),
    Formula(Where(Equal(DirichletCharacter(q, ell, n), Exp(Mul(Mul(Mul(2, Pi), 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, Set(Mod(Pow(5, i), q), For(i), Element(i, ZZGreaterEqual(1))))), Tuple(Tuple(-1, DiscreteLog(Neg(k), 5, q)), Element(k, Set(Mod(Neg(Pow(5, i)), q), For(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, Range(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.

2021-03-15 19:12:00.328586 UTC