Fungrim entry: 921ef0

\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:

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 \gt 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 \gt 0

Definitions:

Fungrim symbol	Notation	Short description
`DedekindEtaEpsilon`	$\varepsilon\!\left(a, b, c, d\right)$	Root of unity in the functional equation of the Dedekind eta function
`Exp`	${e}^{z}$	Exponential function
`ConstPi`	$\pi$	The constant pi (3.14...)
`ConstI`	$i$	Imaginary unit
`DedekindSum`	$s\!\left(n, k\right)$	Dedekind sum
`ZZ`	$\mathbb{Z}$	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))))

Topics using this entry

Dedekind eta function