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MangoldtLambda
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MangoldtLambda

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gives the von Mangoldt function .

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 all

Basic Examples  (2)Summary of the most common use cases

Compute the Mangoldt function at :

In[1]:=1
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https://wolfram.com/xid/0cf1lp1ivlqhhn2-tiwbol
Out[1]=1

Plot the MangoldtLambda sequence for the first 100 numbers:

In[1]:=1
✖
https://wolfram.com/xid/0cf1lp1ivlqhhn2-g5tp8p
Out[1]=1

Scope  (8)Survey of the scope of standard use cases

Numerical Evaluation  (3)

MangoldtLambda works over integers:

In[2]:=2
✖
https://wolfram.com/xid/0cf1lp1ivlqhhn2-ipx2c
Out[2]=2

Compute for large integers:

In[1]:=1
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https://wolfram.com/xid/0cf1lp1ivlqhhn2-i0eqnw
Out[1]=1

MangoldtLambda threads over lists:

In[1]:=1
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https://wolfram.com/xid/0cf1lp1ivlqhhn2-dzq21w
Out[1]=1

Symbolic Manipulation  (5)

TraditionalForm formatting:

In[1]:=1
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https://wolfram.com/xid/0cf1lp1ivlqhhn2-bkedy5

Reduce expressions:

In[1]:=1
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https://wolfram.com/xid/0cf1lp1ivlqhhn2-0no34v
Out[1]=1

Solve equations:

In[1]:=1
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https://wolfram.com/xid/0cf1lp1ivlqhhn2-5dp6eo
Out[1]=1

Sum of MangoldtLambda over divisors:

In[1]:=1
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https://wolfram.com/xid/0cf1lp1ivlqhhn2-v3at3j
Out[1]=1

DirichletTransform of MangoldtLambda:

In[1]:=1
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https://wolfram.com/xid/0cf1lp1ivlqhhn2-b7rnc0
Out[1]=1

Equivalently:

In[2]:=2
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https://wolfram.com/xid/0cf1lp1ivlqhhn2-cabf75
Out[2]=2

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:

In[8]:=8
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https://wolfram.com/xid/0cf1lp1ivlqhhn2-gfysgd
Out[8]=8

Compare MangoldtLambda sequence with logarithm function:

In[1]:=1
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https://wolfram.com/xid/0cf1lp1ivlqhhn2-7dt07t
Out[1]=1

Plot the second Chebyshev function: [more info]

In[1]:=1
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https://wolfram.com/xid/0cf1lp1ivlqhhn2-dw96yo

Demonstrate that it is asymptotic with :

In[2]:=2
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https://wolfram.com/xid/0cf1lp1ivlqhhn2-pak3hs
Out[2]=2

Number Theory  (2)

Use MangoldtLambda to test for a prime power:

In[1]:=1
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https://wolfram.com/xid/0cf1lp1ivlqhhn2-gjn568
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0cf1lp1ivlqhhn2-ilaz74
Out[2]=2

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

In[1]:=1
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https://wolfram.com/xid/0cf1lp1ivlqhhn2-ndgzfg
In[2]:=2
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https://wolfram.com/xid/0cf1lp1ivlqhhn2-5i81mx
Out[2]=2

The more zeros used, the closer the approximation:

In[3]:=3
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https://wolfram.com/xid/0cf1lp1ivlqhhn2-yqh0yo
Out[3]=3

Properties & Relations  (7)Properties of the function, and connections to other functions

MangoldtLambda gives zero except for prime powers:

In[1]:=1
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https://wolfram.com/xid/0cf1lp1ivlqhhn2-g8pqr
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0cf1lp1ivlqhhn2-dm1cu5
Out[2]=2

MangoldtLambda is neither additive or multiplicative:

In[1]:=1
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https://wolfram.com/xid/0cf1lp1ivlqhhn2-mkiwqh
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0cf1lp1ivlqhhn2-d45d4d
Out[2]=2

MangoldtLambda satisfies the identity :

In[1]:=1
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https://wolfram.com/xid/0cf1lp1ivlqhhn2-gphxp8
Out[1]=1

Use MoebiusMu to compute MangoldtLambda:

In[1]:=1
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https://wolfram.com/xid/0cf1lp1ivlqhhn2-dxgm7j
Out[1]=1

Use LCM to compute MangoldtLambda:

In[1]:=1
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https://wolfram.com/xid/0cf1lp1ivlqhhn2-d5ox7
Out[1]=1

Compare with:

In[2]:=2
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https://wolfram.com/xid/0cf1lp1ivlqhhn2-gijtyu
Out[2]=2

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

In[1]:=1
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https://wolfram.com/xid/0cf1lp1ivlqhhn2-89poab
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0cf1lp1ivlqhhn2-6orsme
Out[2]=2

MangoldtLambda satisfies the following identities:

In[1]:=1
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https://wolfram.com/xid/0cf1lp1ivlqhhn2-3bdt0
In[2]:=2
✖
https://wolfram.com/xid/0cf1lp1ivlqhhn2-bq5qw5
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0cf1lp1ivlqhhn2-bt9sps
Out[3]=3
In[4]:=4
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https://wolfram.com/xid/0cf1lp1ivlqhhn2-hez3y
Out[4]=4
In[5]:=5
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https://wolfram.com/xid/0cf1lp1ivlqhhn2-ei5yzd
Out[5]=5

Neat Examples  (3)Surprising or curious use cases

Plot MangoldtLambda for the sum of two squares:

In[1]:=1
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https://wolfram.com/xid/0cf1lp1ivlqhhn2-em4s5g
Out[1]=1

Plot the arguments of the Fourier transform of MangoldtLambda:

In[1]:=1
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https://wolfram.com/xid/0cf1lp1ivlqhhn2-d7etv5
Out[1]=1

Plot the Ulam spiral of MangoldtLambda:

In[1]:=1
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https://wolfram.com/xid/0cf1lp1ivlqhhn2-b6sao8
In[2]:=2
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https://wolfram.com/xid/0cf1lp1ivlqhhn2-tiufn
Out[2]=2
Wolfram Research (2008), MangoldtLambda, Wolfram Language function, https://reference.wolfram.com/language/ref/MangoldtLambda.html.
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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 ]}

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@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 ]}

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@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 ]}