# Fungrim entry: afd27a

$\left|p(n) - \frac{2 \pi}{{\left(24 n - 1\right)}^{3 / 4}} \sum_{k=1}^{N} \frac{A\!\left(n, k\right)}{k} I_{3 / 2}\!\left(\frac{\pi}{k} \sqrt{\frac{2}{3} \left(n - \frac{1}{24}\right)}\right)\right| \le \frac{44 {\pi}^{2}}{225 \sqrt{3 N}} + \frac{\pi \sqrt{2}}{75} \sqrt{\frac{N}{n - 1}} \sinh\!\left(\frac{\pi}{N} \sqrt{\frac{2 n}{3}}\right)$
Assumptions:$n \in \mathbb{Z}_{\ge 2} \;\mathbin{\operatorname{and}}\; N \in \mathbb{Z}_{\ge 1}$
TeX:
\left|p(n) - \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$p(n)$ Integer partition function
Pi$\pi$ The constant pi (3.14...)
Pow${a}^{b}$ Power
Sum$\sum_{n} f(n)$ Sum
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, Pi), Pow(Sub(Mul(24, n), 1), Div(3, 4))), Sum(Mul(Div(HardyRamanujanA(n, k), k), BesselI(Div(3, 2), Mul(Div(Pi, k), Sqrt(Mul(Div(2, 3), Sub(n, Div(1, 24))))))), For(k, 1, N))))), Add(Div(Mul(44, Pow(Pi, 2)), Mul(225, Sqrt(Mul(3, N)))), Mul(Mul(Div(Mul(Pi, Sqrt(2)), 75), Sqrt(Div(N, Sub(n, 1)))), Sinh(Mul(Div(Pi, N), Sqrt(Div(Mul(2, n), 3)))))))),
Variables(n, N),
Assumptions(And(Element(n, ZZGreaterEqual(2)), Element(N, 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.

2020-08-27 09:56:25.682319 UTC