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Fungrim entry: 5adbc3

A ⁣(n,k)=r=0k1δ(gcd(r,k),1)exp ⁣(πi(s ⁣(r,k)2nrk))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:nZ1andkZ1n \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
HardyRamanujanAA ⁣(n,k)A\!\left(n, k\right) Exponential sum in the Hardy-Ramanujan-Rademacher formula
Sumnf ⁣(n)\sum_{n} f\!\left(n\right) Sum
KroneckerDeltaδ(x,y)\delta_{(x,y)} Kronecker delta
GCDgcd ⁣(a,b)\gcd\!\left(a, b\right) Greatest common divisor
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
ZZGreaterEqualZn\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(ConstPi, ConstI), Sub(DedekindSum(r, k), Div(Mul(Mul(2, n), r), k))))), Tuple(r, 0, Sub(k, 1))))),
    Variables(n, k),
    Assumptions(And(Element(n, ZZGreaterEqual(1)), Element(k, ZZGreaterEqual(1)))))

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

2019-08-21 11:44:15.926409 UTC