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Fungrim entry: 921ef0

ε ⁣(a,b,c,d)=exp ⁣(πi(a+d12cs ⁣(d,c)14))\varepsilon\!\left(a, b, c, d\right) = \exp\!\left(\pi i \left(\frac{a + d}{12 c} - s\!\left(d, c\right) - \frac{1}{4}\right)\right)
Assumptions:aZandbZandcZanddZandadbc=1andc>0a \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, b \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, c \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, d \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, a d - b c = 1 \,\mathbin{\operatorname{and}}\, c > 0
TeX:
\varepsilon\!\left(a, b, c, d\right) = \exp\!\left(\pi i \left(\frac{a + d}{12 c} - s\!\left(d, c\right) - \frac{1}{4}\right)\right)

a \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, b \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, c \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, d \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, a d - b c = 1 \,\mathbin{\operatorname{and}}\, c > 0
Definitions:
Fungrim symbol Notation Short description
DedekindEtaEpsilonε ⁣(a,b,c,d)\varepsilon\!\left(a, b, c, d\right) Root of unity in the functional equation of the Dedekind eta function
Expez{e}^{z} Exponential function
ConstPiπ\pi The constant pi (3.14...)
ConstIii Imaginary unit
DedekindSums ⁣(n,k)s\!\left(n, k\right) Dedekind sum
ZZZ\mathbb{Z} Integers
Source code for this entry:
Entry(ID("921ef0"),
    Formula(Equal(DedekindEtaEpsilon(a, b, c, d), Exp(Mul(Mul(ConstPi, ConstI), Sub(Sub(Div(Add(a, d), Mul(12, c)), DedekindSum(d, c)), Div(1, 4)))))),
    Variables(a, b, c, d),
    Assumptions(And(Element(a, ZZ), Element(b, ZZ), Element(c, ZZ), Element(d, ZZ), Equal(Sub(Mul(a, d), Mul(b, c)), 1), Greater(c, 0))))

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2019-08-19 14:38:23.809000 UTC