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# Fungrim entry: 831ea4

$\Lambda = \mathop{\operatorname{zero*}\,}\limits_{x \in \left(0, 1\right)} \left[-\frac{1}{8} + \sum_{n=1}^{\infty} \left|\sum_{d \mid n} {\left(-1\right)}^{d} d\right| {x}^{n}\right]$
TeX:
\Lambda = \mathop{\operatorname{zero*}\,}\limits_{x \in \left(0, 1\right)} \left[-\frac{1}{8} + \sum_{n=1}^{\infty} \left|\sum_{d \mid n} {\left(-1\right)}^{d} d\right| {x}^{n}\right]
Definitions:
Fungrim symbol Notation Short description
HalphenConstant$\Lambda$ Halphen's constant (one-ninth constant) 0.10765...
UniqueZero$\mathop{\operatorname{zero*}\,}\limits_{x \in S} f(x)$ Unique zero (root) of function
Sum$\sum_{n} f(n)$ Sum
Abs$\left|z\right|$ Absolute value
DivisorSum$\sum_{k \mid n} f(k)$ Sum over divisors
Pow${a}^{b}$ Power
Infinity$\infty$ Positive infinity
OpenInterval$\left(a, b\right)$ Open interval
Source code for this entry:
Entry(ID("831ea4"),
Formula(Equal(HalphenConstant, UniqueZero(Add(Neg(Div(1, 8)), Sum(Mul(Abs(DivisorSum(Mul(Pow(-1, d), d), For(d, n))), Pow(x, n)), For(n, 1, Infinity))), ForElement(x, OpenInterval(0, 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