Fungrim entry: 5adbc3

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

Partition function