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

Halphen's constant