Fungrim entry: afd27a

\left|p\!\left(n\right) - \frac{2 \pi}{{\left(24 n - 1\right)}^{3 / 4}} \sum_{k=1}^{N} \frac{A\!\left(n, k\right)}{k} I_{3 / 2}\!\left(\frac{\pi}{k} \sqrt{\frac{2}{3} \left(n - \frac{1}{24}\right)}\right)\right| \le \frac{44 {\pi}^{2}}{225 \sqrt{3 N}} + \frac{\pi \sqrt{2}}{75} \sqrt{\frac{N}{n - 1}} \sinh\!\left(\frac{\pi}{N} \sqrt{\frac{2 n}{3}}\right)

Assumptions:

n \in \mathbb{Z}_{\ge 2} \,\mathbin{\operatorname{and}}\, N \in \mathbb{Z}_{\ge 1}

TeX:

\left|p\!\left(n\right) - \frac{2 \pi}{{\left(24 n - 1\right)}^{3 / 4}} \sum_{k=1}^{N} \frac{A\!\left(n, k\right)}{k} I_{3 / 2}\!\left(\frac{\pi}{k} \sqrt{\frac{2}{3} \left(n - \frac{1}{24}\right)}\right)\right| \le \frac{44 {\pi}^{2}}{225 \sqrt{3 N}} + \frac{\pi \sqrt{2}}{75} \sqrt{\frac{N}{n - 1}} \sinh\!\left(\frac{\pi}{N} \sqrt{\frac{2 n}{3}}\right)

n \in \mathbb{Z}_{\ge 2} \,\mathbin{\operatorname{and}}\, N \in \mathbb{Z}_{\ge 1}

Definitions:

Fungrim symbol	Notation	Short description
`Abs`	$\left\|z\right\|$	Absolute value
`PartitionsP`	$p\!\left(n\right)$	Integer partition function
`ConstPi`	$\pi$	The constant pi (3.14...)
`Pow`	${a}^{b}$	Power
`HardyRamanujanA`	$A\!\left(n, k\right)$	Exponential sum in the Hardy-Ramanujan-Rademacher formula
`BesselI`	$I_{\nu}\!\left(z\right)$	Modified Bessel function of the first kind
`Sqrt`	$\sqrt{z}$	Principal square root
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("afd27a"),
    Formula(LessEqual(Abs(Sub(PartitionsP(n), Mul(Div(Mul(2, ConstPi), Pow(Sub(Mul(24, n), 1), Div(3, 4))), Sum(Mul(Div(HardyRamanujanA(n, k), k), BesselI(Div(3, 2), Mul(Div(ConstPi, k), Sqrt(Mul(Div(2, 3), Sub(n, Div(1, 24))))))), Tuple(k, 1, N))))), Add(Div(Mul(44, Pow(ConstPi, 2)), Mul(225, Sqrt(Mul(3, N)))), Mul(Mul(Div(Mul(ConstPi, Sqrt(2)), 75), Sqrt(Div(N, Sub(n, 1)))), Sinh(Mul(Div(ConstPi, N), Sqrt(Div(Mul(2, n), 3)))))))),
    Variables(n, N),
    Assumptions(And(Element(n, ZZGreaterEqual(2)), Element(N, ZZGreaterEqual(1)))))

Topics using this entry

Partition function