# 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 Manipulation(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 blue, 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:

Introduced in 2008
(7.0)