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RamanujanTau
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RamanujanTau

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gives the Ramanujan function .

Details

  • Integer mathematical function.
  • gives the coefficient of in the series expansion of .
  • RamanujanTau automatically threads over lists.

Examples

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

The first 10 values of RamanujanTau:

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In[1]:=1
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https://wolfram.com/xid/0ixg9i81elravql-nr1
Direct link to example
Out[1]=1

Plot over a subset of the reals:

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In[1]:=1
✖
https://wolfram.com/xid/0ixg9i81elravql-puhw05
Direct link to example
Out[1]=1

Scope  (12)Survey of the scope of standard use cases

Numerical Evaluation  (3)

Evaluate numerically:

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In[1]:=1
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https://wolfram.com/xid/0ixg9i81elravql-l274ju
Direct link to example
Out[1]=1
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In[2]:=2
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https://wolfram.com/xid/0ixg9i81elravql-cksbl4
Direct link to example
Out[2]=2

Evaluate efficiently for large values of the argument:

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In[1]:=1
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https://wolfram.com/xid/0ixg9i81elravql-jy8ead
Direct link to example
Out[1]=1
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In[2]:=2
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https://wolfram.com/xid/0ixg9i81elravql-fgq19x
Direct link to example
Out[2]=2

Compute the elementwise values of an array:

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In[1]:=1
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https://wolfram.com/xid/0ixg9i81elravql-thgd2
Direct link to example
Out[1]=1

Or compute the matrix RamanujanTau function using MatrixFunction:

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In[2]:=2
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https://wolfram.com/xid/0ixg9i81elravql-o5jpo
Direct link to example
Out[2]=2

Specific Values  (2)

Values at fixed points:

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In[1]:=1
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https://wolfram.com/xid/0ixg9i81elravql-nww7l
Direct link to example
Out[1]=1

Value at zero:

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In[1]:=1
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https://wolfram.com/xid/0ixg9i81elravql-bmqd0y
Direct link to example
Out[1]=1

Visualization  (3)

Plot the RamanujanTau function:

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In[1]:=1
✖
https://wolfram.com/xid/0ixg9i81elravql-b1j98m
Direct link to example
Out[1]=1

Plot the contours of the RamanujanTau function:

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In[1]:=1
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https://wolfram.com/xid/0ixg9i81elravql-j7or69
Direct link to example
Out[1]=1

Plot the RamanujanTau function in three dimensions:

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In[1]:=1
✖
https://wolfram.com/xid/0ixg9i81elravql-f6wno4
Direct link to example
Out[1]=1

Function Properties  (4)

RamanujanTau is only defined for integer inputs:

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In[1]:=1
✖
https://wolfram.com/xid/0ixg9i81elravql-cl7ele
Direct link to example
Out[1]=1
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In[2]:=2
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https://wolfram.com/xid/0ixg9i81elravql-de3irc
Direct link to example
Out[2]=2

RamanujanTau threads over lists:

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In[1]:=1
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https://wolfram.com/xid/0ixg9i81elravql-bmhpyp
Direct link to example
Out[1]=1

RamanujanTauL is everywhere singular:

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In[1]:=1
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https://wolfram.com/xid/0ixg9i81elravql-mdtl3h
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
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https://wolfram.com/xid/0ixg9i81elravql-mn5jws
Direct link to example
Out[2]=2

Traditional form:

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In[1]:=1
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https://wolfram.com/xid/0ixg9i81elravql-22ky2r
Direct link to example

Applications  (7)Sample problems that can be solved with this function

Logarithmic plot of RamanujanTau:

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In[1]:=1
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https://wolfram.com/xid/0ixg9i81elravql-tg5j2g
Direct link to example
Out[1]=1

The first prime value of RamanujanTau:

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In[1]:=1
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https://wolfram.com/xid/0ixg9i81elravql-moes1d
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0ixg9i81elravql-c8nxx4
Direct link to example
Out[2]=2

The first 20,000 values are nonzero, satisfying Lehmer's conjecture [more info]:

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In[1]:=1
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https://wolfram.com/xid/0ixg9i81elravql-mv5v8t
Direct link to example
Out[1]=1

Plot of at primes :

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In[1]:=1
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https://wolfram.com/xid/0ixg9i81elravql-5zy7vo
Direct link to example
Out[1]=1

The modular discriminant:

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In[1]:=1
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https://wolfram.com/xid/0ixg9i81elravql-8v5zw0
Direct link to example

Relation with DedekindEta:

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In[2]:=2
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https://wolfram.com/xid/0ixg9i81elravql-mwacl5
Direct link to example
Out[2]=2

The summatory -function [more info]:

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In[1]:=1
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https://wolfram.com/xid/0ixg9i81elravql-i9lhlf
Direct link to example
Out[1]=1

The modular discriminant:

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In[1]:=1
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https://wolfram.com/xid/0ixg9i81elravql-oghhf
Direct link to example

Relation with DedekindEta:

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In[2]:=2
✖
https://wolfram.com/xid/0ixg9i81elravql-fmgnqi
Direct link to example
Out[2]=2

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

The first 10 values of RamanujanTau using Product:

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In[1]:=1
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https://wolfram.com/xid/0ixg9i81elravql-kti93k
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0ixg9i81elravql-247f03
Direct link to example
Out[2]=2

RamanujanTau is multiplicative for coprime integers:

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In[1]:=1
✖
https://wolfram.com/xid/0ixg9i81elravql-6itnhx
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0ixg9i81elravql-gjuu2u
Direct link to example
Out[2]=2

For prime :

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In[1]:=1
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https://wolfram.com/xid/0ixg9i81elravql-x3uc9j
Direct link to example
Out[1]=1

Congruence relations:

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In[1]:=1
✖
https://wolfram.com/xid/0ixg9i81elravql-5b4dco
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0ixg9i81elravql-zv00l
Direct link to example
Out[2]=2
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In[3]:=3
✖
https://wolfram.com/xid/0ixg9i81elravql-06zjot
Direct link to example
Out[3]=3

Representation of an integer as the sum of 24 squares:

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In[1]:=1
✖
https://wolfram.com/xid/0ixg9i81elravql-gzcyoa
Direct link to example
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0ixg9i81elravql-9kvnwa
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0ixg9i81elravql-lxyc0k
Direct link to example
Out[3]=3

RamanujanTauL is the Dirichlet -function associated with RamanujanTau:

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In[1]:=1
✖
https://wolfram.com/xid/0ixg9i81elravql-k4xjoo
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0ixg9i81elravql-n5namr
Direct link to example
Out[2]=2

FindSequenceFunction can recognize the RamanujanTau sequence:

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In[1]:=1
✖
https://wolfram.com/xid/0ixg9i81elravql-hj2mn6
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0ixg9i81elravql-5okec
Direct link to example
Out[2]=2

Possible Issues  (1)Common pitfalls and unexpected behavior

Large prime numbers can take a long time:

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In[1]:=1
✖
https://wolfram.com/xid/0ixg9i81elravql-j8g6qe
Direct link to example
Out[1]=1

Neat Examples  (3)Surprising or curious use cases

Successive differences of RamanujanTau modulo 3:

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In[1]:=1
✖
https://wolfram.com/xid/0ixg9i81elravql-j89uev
Direct link to example
Out[1]=1

A representation of zero in terms of RamanujanTau:

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In[1]:=1
✖
https://wolfram.com/xid/0ixg9i81elravql-jx4f67
Direct link to example
Out[1]=1

Find digit counts for RamanujanTau[10^12]:

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In[1]:=1
✖
https://wolfram.com/xid/0ixg9i81elravql-l0tj70
Direct link to example
Out[1]=1
Wolfram Research (2007), RamanujanTau, Wolfram Language function, https://reference.wolfram.com/language/ref/RamanujanTau.html.
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Wolfram Research (2007), RamanujanTau, Wolfram Language function, https://reference.wolfram.com/language/ref/RamanujanTau.html.

Text

Wolfram Research (2007), RamanujanTau, Wolfram Language function, https://reference.wolfram.com/language/ref/RamanujanTau.html.

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Wolfram Research (2007), RamanujanTau, Wolfram Language function, https://reference.wolfram.com/language/ref/RamanujanTau.html.

CMS

Wolfram Language. 2007. "RamanujanTau." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/RamanujanTau.html.

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Wolfram Language. 2007. "RamanujanTau." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/RamanujanTau.html.

APA

Wolfram Language. (2007). RamanujanTau. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/RamanujanTau.html

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Wolfram Language. (2007). RamanujanTau. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/RamanujanTau.html

BibTeX

@misc{reference.wolfram_2025_ramanujantau, author="Wolfram Research", title="{RamanujanTau}", year="2007", howpublished="\url{https://reference.wolfram.com/language/ref/RamanujanTau.html}", note=[Accessed: 26-March-2025 ]}

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@misc{reference.wolfram_2025_ramanujantau, author="Wolfram Research", title="{RamanujanTau}", year="2007", howpublished="\url{https://reference.wolfram.com/language/ref/RamanujanTau.html}", note=[Accessed: 26-March-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_ramanujantau, organization={Wolfram Research}, title={RamanujanTau}, year={2007}, url={https://reference.wolfram.com/language/ref/RamanujanTau.html}, note=[Accessed: 26-March-2025 ]}

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@online{reference.wolfram_2025_ramanujantau, organization={Wolfram Research}, title={RamanujanTau}, year={2007}, url={https://reference.wolfram.com/language/ref/RamanujanTau.html}, note=[Accessed: 26-March-2025 ]}