MangoldtLambda

The von Mangoldt function: $\ln p$ if $n$ is a power of the prime $p$, else 0.

✓ checked against that system · ! checked, with disagreements (hover a chip)Wolfram LanguageMangoldtLambda✓↗

MangoldtLambda(n)$\ln p$ for $n=p^k$, else 0

Typed signatures

Type	Package	Applies
(integer) -> number	number-theory	always

Domain: Number theory

Details

• Exact: the result is a symbolic $\ln p$, not a numeric approximation — wrap in N to get one.
• $\sum_{d\mid n}\Lambda(d)=\ln n$; central to the prime-counting proofs (Chebyshev's $\psi$ and $\theta$ functions).

Examples

2 more kept as data, checked against the oracles

Basic #

$8=2^3$

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In$\mathrm{MangoldtLambda}(8)$Out$\ln(2)$

6 is not a prime power

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In$\mathrm{MangoldtLambda}(6)$Out$0$

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In$\mathrm{MangoldtLambda}(7)$Out$\ln(7)$

A prime power $5^2$ gives $\ln 5$

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In$\mathrm{MangoldtLambda}(25)$Out$\ln(5)$

Scope #

Numerically, $\ln3$

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In$N(\mathrm{MangoldtLambda}(9))$Out$1.098\,612\,288\,668\,109\,8$

Listable

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In$\mathrm{MangoldtLambda}(\bigl\lbrack2, 4, 6, 25\bigr\rbrack)$Out$\bigl\lbrack\ln(2), \ln(2), 0, \ln(5)\bigr\rbrack$

Any power of 5 gives the same $\ln 5$

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In$\mathrm{MangoldtLambda}(125)$Out$\ln(5)$

Huge prime powers are recognised without factoring

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In$\mathrm{MangoldtLambda}(7^{300})$Out$\ln(7)$

Threads over a list; non-prime-powers give $0$

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In$\mathrm{MangoldtLambda}(\bigl\lbrack2, 6, 9, 8, 15\bigr\rbrack)$Out$\bigl\lbrack\ln(2), 0, \ln(3), \ln(2), 0\bigr\rbrack$

Applications #

$81=3^4$ is a prime power, so the value is nonzero

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In$\mathrm{MangoldtLambda}(81)≠0$Out$\mathrm{True}$

Properties #

Not additive: $\Lambda(6)=0\ne\Lambda(2)+\Lambda(3)$

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In$\bigl\lbrack\mathrm{MangoldtLambda}(6), \mathrm{MangoldtLambda}(2)+\mathrm{MangoldtLambda}(3)\bigr\rbrack$Out$\bigl\lbrack0, \ln(2)+\ln(3)\bigr\rbrack$

Not multiplicative either: $\Lambda(6)=0$

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In$\bigl\lbrack\mathrm{MangoldtLambda}(6), \mathrm{MangoldtLambda}(2)\mathrm{MangoldtLambda}(3)\bigr\rbrack$Out$\bigl\lbrack0, \ln(2)\ln(3)\bigr\rbrack$

A prime gives its logarithm

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In$\mathrm{MangoldtLambda}(5)$Out$\ln(5)$

$\Lambda(2)=\ln2$

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In$\mathrm{MangoldtLambda}(2)$Out$\ln(2)$

Known value · Wolfram MathematicalFunctionData, MangoldtLambda

$\Lambda(3)=\ln3$

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In$\mathrm{MangoldtLambda}(3)$Out$\ln(3)$

Known value · Wolfram MathematicalFunctionData, MangoldtLambda

$4=2^2$, so $\ln 2$

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In$\mathrm{MangoldtLambda}(4)$Out$\ln(2)$

Known value · Wolfram MathematicalFunctionData, MangoldtLambda

Two distinct primes give $0$

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In$\mathrm{MangoldtLambda}(10)$Out$0$

Known value · Wolfram MathematicalFunctionData, MangoldtLambda

Prime, so $\ln 11$

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In$\mathrm{MangoldtLambda}(11)$Out$\ln(11)$

Known value · Wolfram MathematicalFunctionData, MangoldtLambda

$12=2^2\cdot3$ is not a prime power

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In$\mathrm{MangoldtLambda}(12)$Out$0$

Known value · Wolfram MathematicalFunctionData, MangoldtLambda

$\sum_{n\le20}\Lambda(n)=\ln\operatorname{lcm}(1..20)$

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In$\mathrm{Simplify}(\sum_{n=20}\mathrm{MangoldtLambda}(n))$Out$2\ln(3)+4\ln(2)+\ln(1\,616\,615)$

Prime, so $\ln 13$

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In$\mathrm{MangoldtLambda}(13)$Out$\ln(13)$

Known value · Wolfram MathematicalFunctionData, MangoldtLambda

A semiprime gives $0$

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In$\mathrm{MangoldtLambda}(14)$Out$0$

Known value · Wolfram MathematicalFunctionData, MangoldtLambda

Product of two primes: $0$

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In$\mathrm{MangoldtLambda}(15)$Out$0$

Known value · Wolfram MathematicalFunctionData, MangoldtLambda

$2^4$ gives $\ln2$, not $4\ln2$

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In$\mathrm{MangoldtLambda}(16)$Out$\ln(2)$

Known value · Wolfram MathematicalFunctionData, MangoldtLambda

Prime, so $\ln 17$

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In$\mathrm{MangoldtLambda}(17)$Out$\ln(17)$

Known value · Wolfram MathematicalFunctionData, MangoldtLambda

$\sum_{d\mid n}\Lambda(d)=\ln n$, shown at $n=16$

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In$\ln(16)=\mathrm{DivisorSum}(16, ()↦\mathrm{MangoldtLambda}(\operatorname{\_1}))$Out$\mathrm{True}$

Possible issues #

$\Lambda(1)=0$

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In$\mathrm{MangoldtLambda}(1)$Out$0$

About

Examples	29
Checked against	Wolfram Language 29 agree
Crosswalk	Wolfram Language

See also: IsPrimePower, FactorInteger