$A\!\left(n, k\right) = \sum_{r=0}^{k - 1} \delta_{(\gcd\left(r, k\right),1)} \exp\!\left(\pi i \left(s\!\left(r, k\right) - \frac{2 n r}{k}\right)\right)$
Assumptions:$n \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; k \in \mathbb{Z}_{\ge 1}$
TeX:
A\!\left(n, k\right) = \sum_{r=0}^{k - 1} \delta_{(\gcd\left(r, k\right),1)} \exp\!\left(\pi i \left(s\!\left(r, k\right) - \frac{2 n r}{k}\right)\right)

n \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; k \in \mathbb{Z}_{\ge 1}
Definitions:
Fungrim symbol Notation Short description
HardyRamanujanA$A\!\left(n, k\right)$ Exponential sum in the Hardy-Ramanujan-Rademacher formula
Sum$\sum_{n} f(n)$ Sum
KroneckerDelta$\delta_{(x,y)}$ Kronecker delta
GCD$\gcd\!\left(a, b\right)$ Greatest common divisor
Exp${e}^{z}$ Exponential function
Pi$\pi$ The constant pi (3.14...)
ConstI$i$ Imaginary unit
DedekindSum$s\!\left(n, k\right)$ Dedekind sum
ZZGreaterEqual$\mathbb{Z}_{\ge n}$ Integers greater than or equal to n
Source code for this entry:
Entry(ID("5adbc3"),
Formula(Equal(HardyRamanujanA(n, k), Sum(Mul(KroneckerDelta(GCD(r, k), 1), Exp(Mul(Mul(Pi, ConstI), Sub(DedekindSum(r, k), Div(Mul(Mul(2, n), r), k))))), For(r, 0, Sub(k, 1))))),
Variables(n, k),
Assumptions(And(Element(n, ZZGreaterEqual(1)), Element(k, ZZGreaterEqual(1)))))

## Topics using this entry

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