MangoldtLambda
✖
MangoldtLambda
Details

- MangoldtLambda is also know as von Mangoldt function.
- Integer mathematical function, suitable for both symbolic and numerical manipulation.
- MangoldtLambda[n] 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, MangoldtLambda[n] returns 0 unless m is equal to 1, in which case it gives Log[p1].

Examples
open allclose allBasic Examples (2)Summary of the most common use cases
Compute the Mangoldt function at :

https://wolfram.com/xid/0cf1lp1ivlqhhn2-tiwbol

Plot the MangoldtLambda sequence for the first 100 numbers:

https://wolfram.com/xid/0cf1lp1ivlqhhn2-g5tp8p

Scope (8)Survey of the scope of standard use cases
Numerical Evaluation (3)
MangoldtLambda works over integers:

https://wolfram.com/xid/0cf1lp1ivlqhhn2-ipx2c


https://wolfram.com/xid/0cf1lp1ivlqhhn2-i0eqnw

MangoldtLambda threads over lists:

https://wolfram.com/xid/0cf1lp1ivlqhhn2-dzq21w

Symbolic Manipulation (5)
TraditionalForm formatting:

https://wolfram.com/xid/0cf1lp1ivlqhhn2-bkedy5


https://wolfram.com/xid/0cf1lp1ivlqhhn2-0no34v


https://wolfram.com/xid/0cf1lp1ivlqhhn2-5dp6eo

Sum of MangoldtLambda over divisors:

https://wolfram.com/xid/0cf1lp1ivlqhhn2-v3at3j

DirichletTransform of MangoldtLambda:

https://wolfram.com/xid/0cf1lp1ivlqhhn2-b7rnc0


https://wolfram.com/xid/0cf1lp1ivlqhhn2-cabf75

Applications (5)Sample problems that can be solved with this function
Basic Applications (3)
Highlight numbers n for which in black, and the prime bases of numbers n for which
in red:

https://wolfram.com/xid/0cf1lp1ivlqhhn2-gfysgd

Compare MangoldtLambda sequence with logarithm function:

https://wolfram.com/xid/0cf1lp1ivlqhhn2-7dt07t

Plot the second Chebyshev function: [more info]

https://wolfram.com/xid/0cf1lp1ivlqhhn2-dw96yo
Demonstrate that it is asymptotic with :

https://wolfram.com/xid/0cf1lp1ivlqhhn2-pak3hs

Number Theory (2)
Use MangoldtLambda to test for a prime power:

https://wolfram.com/xid/0cf1lp1ivlqhhn2-gjn568


https://wolfram.com/xid/0cf1lp1ivlqhhn2-ilaz74

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

https://wolfram.com/xid/0cf1lp1ivlqhhn2-ndgzfg

https://wolfram.com/xid/0cf1lp1ivlqhhn2-5i81mx

The more zeros used, the closer the approximation:

https://wolfram.com/xid/0cf1lp1ivlqhhn2-yqh0yo

Properties & Relations (7)Properties of the function, and connections to other functions
MangoldtLambda gives zero except for prime powers:

https://wolfram.com/xid/0cf1lp1ivlqhhn2-g8pqr


https://wolfram.com/xid/0cf1lp1ivlqhhn2-dm1cu5

MangoldtLambda is neither additive or multiplicative:

https://wolfram.com/xid/0cf1lp1ivlqhhn2-mkiwqh


https://wolfram.com/xid/0cf1lp1ivlqhhn2-d45d4d

MangoldtLambda satisfies the identity :

https://wolfram.com/xid/0cf1lp1ivlqhhn2-gphxp8

Use MoebiusMu to compute MangoldtLambda:

https://wolfram.com/xid/0cf1lp1ivlqhhn2-dxgm7j

Use LCM to compute MangoldtLambda:

https://wolfram.com/xid/0cf1lp1ivlqhhn2-d5ox7


https://wolfram.com/xid/0cf1lp1ivlqhhn2-gijtyu

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

https://wolfram.com/xid/0cf1lp1ivlqhhn2-89poab


https://wolfram.com/xid/0cf1lp1ivlqhhn2-6orsme

MangoldtLambda satisfies the following identities:

https://wolfram.com/xid/0cf1lp1ivlqhhn2-3bdt0

https://wolfram.com/xid/0cf1lp1ivlqhhn2-bq5qw5


https://wolfram.com/xid/0cf1lp1ivlqhhn2-bt9sps


https://wolfram.com/xid/0cf1lp1ivlqhhn2-hez3y


https://wolfram.com/xid/0cf1lp1ivlqhhn2-ei5yzd

Neat Examples (3)Surprising or curious use cases
Plot MangoldtLambda for the sum of two squares:

https://wolfram.com/xid/0cf1lp1ivlqhhn2-em4s5g

Plot the arguments of the Fourier transform of MangoldtLambda:

https://wolfram.com/xid/0cf1lp1ivlqhhn2-d7etv5

Plot the Ulam spiral of MangoldtLambda:

https://wolfram.com/xid/0cf1lp1ivlqhhn2-b6sao8

https://wolfram.com/xid/0cf1lp1ivlqhhn2-tiufn

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.
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.
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
Wolfram Language. (2008). MangoldtLambda. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/MangoldtLambda.html
BibTeX
@misc{reference.wolfram_2025_mangoldtlambda, author="Wolfram Research", title="{MangoldtLambda}", year="2008", howpublished="\url{https://reference.wolfram.com/language/ref/MangoldtLambda.html}", note=[Accessed: 09-May-2025
]}
BibLaTeX
@online{reference.wolfram_2025_mangoldtlambda, organization={Wolfram Research}, title={MangoldtLambda}, year={2008}, url={https://reference.wolfram.com/language/ref/MangoldtLambda.html}, note=[Accessed: 09-May-2025
]}