# MangoldtLambda

gives the von Mangoldt function .

# Details

• MangoldtLambda is also know as von Mangoldt function.
• Integer mathematical function, suitable for both symbolic and numerical manipulation.
• gives zero unless n is a prime power, in which case it gives the logarithm of the prime.
• For a positive integer n= p1k1 pmkm with pi primes, returns 0 unless m is equal to 1, in which case it gives Log[p1].

# Examples

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## Basic Examples(2)

Compute the Mangoldt function at :

Plot the MangoldtLambda sequence for the first 100 numbers:

## Scope(8)

### Numerical Evaluation(3)

MangoldtLambda works over integers:

Compute for large integers:

### Symbolic Manipulation(5)

Reduce expressions:

Solve equations:

Sum of MangoldtLambda over divisors:

Equivalently:

## Applications(5)

### Basic Applications(3)

Highlight numbers n for which in black, and the prime bases of numbers n for which in red:

Compare MangoldtLambda sequence with logarithm function:

Demonstrate that it is asymptotic with :

### Number Theory(2)

Use MangoldtLambda to test for a prime power:

Plot an approximation of the number of primes and prime powers using MangoldtLambda and ZetaZero:

The more zeros used, the closer the approximation:

## Properties & Relations(7)

MangoldtLambda gives zero except for prime powers:

MangoldtLambda is neither additive or multiplicative:

MangoldtLambda satisfies the identity :

Use MoebiusMu to compute MangoldtLambda:

Use LCM to compute MangoldtLambda:

Compare with:

The sum of MangoldtLambda of the first n integers is equal to the natural log of the LCM of the first n integers:

MangoldtLambda satisfies the following identities:

## Neat Examples(3)

Plot MangoldtLambda for the sum of two squares:

Plot the arguments of the Fourier transform of MangoldtLambda:

Plot the Ulam spiral of MangoldtLambda:

Wolfram Research (2008), MangoldtLambda, Wolfram Language function, https://reference.wolfram.com/language/ref/MangoldtLambda.html.

#### Text

Wolfram Research (2008), MangoldtLambda, Wolfram Language function, https://reference.wolfram.com/language/ref/MangoldtLambda.html.

#### CMS

Wolfram Language. 2008. "MangoldtLambda." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/MangoldtLambda.html.

#### APA

Wolfram Language. (2008). MangoldtLambda. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/MangoldtLambda.html

#### BibTeX

@misc{reference.wolfram_2024_mangoldtlambda, author="Wolfram Research", title="{MangoldtLambda}", year="2008", howpublished="\url{https://reference.wolfram.com/language/ref/MangoldtLambda.html}", note=[Accessed: 13-August-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_mangoldtlambda, organization={Wolfram Research}, title={MangoldtLambda}, year={2008}, url={https://reference.wolfram.com/language/ref/MangoldtLambda.html}, note=[Accessed: 13-August-2024 ]}