# 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:$a \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; b \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; c \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; d \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; a d - b c = 1 \;\mathbin{\operatorname{and}}\; c > 0$
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
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
Pi$\pi$ 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(Pi, 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

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

2020-08-27 09:56:25.682319 UTC