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RamanujanTauL
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RamanujanTauL

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gives the Ramanujan tau Dirichlet L-function .

Details

  • Mathematical function, suitable for both symbolic and numerical manipulation.
  • For Re(s)>6, is given by the Dirichlet series , where is the Ramanujan function.
  • RamanujanTauL can be evaluated to arbitrary numerical precision.
  • RamanujanTauL automatically threads over lists.

Examples

open allclose all

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

Evaluate numerically:

In[2]:=2
✖
https://wolfram.com/xid/0pniox72hyojcfab-tkr
Out[2]=2

Plot over a subset of the reals:

In[1]:=1
✖
https://wolfram.com/xid/0pniox72hyojcfab-dx9l95
Out[1]=1

Scope  (20)Survey of the scope of standard use cases

Numerical Evaluation  (6)

Evaluate numerically:

In[1]:=1
✖
https://wolfram.com/xid/0pniox72hyojcfab-l274ju
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0pniox72hyojcfab-pmo0yc
Out[2]=2

Evaluate to high precision:

In[1]:=1
✖
https://wolfram.com/xid/0pniox72hyojcfab-b0wt9
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0pniox72hyojcfab-zn1q5
Out[2]=2

The precision of the output tracks the precision of the input:

In[3]:=3
✖
https://wolfram.com/xid/0pniox72hyojcfab-y7k4a
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0pniox72hyojcfab-bj7l9h
Out[4]=4

Complex number inputs:

In[1]:=1
✖
https://wolfram.com/xid/0pniox72hyojcfab-hfml09
Out[1]=1

Evaluate efficiently at high precision:

In[1]:=1
✖
https://wolfram.com/xid/0pniox72hyojcfab-di5gcr
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0pniox72hyojcfab-bq2c6r
Out[2]=2

Compute average-case statistical intervals using Around:

In[1]:=1
✖
https://wolfram.com/xid/0pniox72hyojcfab-cw18bq
Out[1]=1

Compute the elementwise values of an array:

In[1]:=1
✖
https://wolfram.com/xid/0pniox72hyojcfab-thgd2
Out[1]=1

Or compute the matrix RamanujanTauL function using MatrixFunction:

In[2]:=2
✖
https://wolfram.com/xid/0pniox72hyojcfab-o5jpo
Out[2]=2

Specific Values  (2)

Value at zero:

In[1]:=1
✖
https://wolfram.com/xid/0pniox72hyojcfab-cgkwdk
Out[1]=1

Find a value of x for which RamanujanTauL[x]=0.8:

In[1]:=1
✖
https://wolfram.com/xid/0pniox72hyojcfab-f2hrld
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0pniox72hyojcfab-f1imhf
Out[2]=2

Visualization  (2)

Plot the RamanujanTauL:

In[1]:=1
✖
https://wolfram.com/xid/0pniox72hyojcfab-b1j98m
Out[1]=1

Plot the real part of RamanujanTauL function:

In[1]:=1
✖
https://wolfram.com/xid/0pniox72hyojcfab-bxaed9
Out[1]=1

Plot the imaginary part of RamanujanTauL function:

In[2]:=2
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https://wolfram.com/xid/0pniox72hyojcfab-eweler
Out[2]=2

Function Properties  (10)

RamanujanTauL is defined for all real values:

In[1]:=1
✖
https://wolfram.com/xid/0pniox72hyojcfab-cl7ele
Out[1]=1

Complex domain:

In[2]:=2
✖
https://wolfram.com/xid/0pniox72hyojcfab-de3irc
Out[2]=2

Bounds on the function range of RamanujanTauL:

In[1]:=1
✖
https://wolfram.com/xid/0pniox72hyojcfab-evf2yr
Out[1]=1

RamanujanTauL threads over lists:

In[1]:=1
✖
https://wolfram.com/xid/0pniox72hyojcfab-dae6v5
Out[1]=1

RamanujanTauL is an analytic function of x:

In[1]:=1
✖
https://wolfram.com/xid/0pniox72hyojcfab-h5x4l2
Out[1]=1

RamanujanTauL is neither non-increasing nor non-decreasing:

In[1]:=1
✖
https://wolfram.com/xid/0pniox72hyojcfab-g6kynf
Out[1]=1

RamanujanTauL is not injective:

In[1]:=1
✖
https://wolfram.com/xid/0pniox72hyojcfab-gi38d7
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0pniox72hyojcfab-ctca0g
Out[2]=2

RamanujanTauL is surjective:

In[1]:=1
✖
https://wolfram.com/xid/0pniox72hyojcfab-hkqec4
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0pniox72hyojcfab-hdm869
Out[2]=2

RamanujanTauL is neither non-negative nor non-positive:

In[1]:=1
✖
https://wolfram.com/xid/0pniox72hyojcfab-84dui
Out[1]=1

RamanujanTauL has no singularities or discontinuities:

In[1]:=1
✖
https://wolfram.com/xid/0pniox72hyojcfab-mdtl3h
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0pniox72hyojcfab-mn5jws
Out[2]=2

RamanujanTauL is neither convex nor concave:

In[1]:=1
✖
https://wolfram.com/xid/0pniox72hyojcfab-kdss3
Out[1]=1

Applications  (5)Sample problems that can be solved with this function

Plot on the critical line:

In[1]:=1
✖
https://wolfram.com/xid/0pniox72hyojcfab-gqi0gc
Out[1]=1

Find a zero of RamanujanTauL:

In[1]:=1
✖
https://wolfram.com/xid/0pniox72hyojcfab-p6km1v
Out[1]=1

The number of zeros on the critical strip from 0 to :

In[1]:=1
✖
https://wolfram.com/xid/0pniox72hyojcfab-j7pggl
In[2]:=2
✖
https://wolfram.com/xid/0pniox72hyojcfab-8reysz
Out[2]=2

Plot of the real part:

In[1]:=1
✖
https://wolfram.com/xid/0pniox72hyojcfab-038fi2
Out[1]=1

Define and visualize the Ramanujan function:

In[1]:=1
✖
https://wolfram.com/xid/0pniox72hyojcfab-2jev29
In[2]:=2
✖
https://wolfram.com/xid/0pniox72hyojcfab-p5jkn1
Out[2]=2

Properties & Relations  (5)Properties of the function, and connections to other functions

Functional equation:

In[1]:=1
✖
https://wolfram.com/xid/0pniox72hyojcfab-yudygy
In[2]:=2
✖
https://wolfram.com/xid/0pniox72hyojcfab-11jt4n
Out[2]=2

Approximate RamanujanTauL using an Euler product formula:

In[1]:=1
✖
https://wolfram.com/xid/0pniox72hyojcfab-kprh1b
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0pniox72hyojcfab-whydbe
Out[2]=2

On the critical line, RamanujanTauL can be split into RamanujanTauTheta and RamanujanTauZ:

In[1]:=1
✖
https://wolfram.com/xid/0pniox72hyojcfab-70byrp
Out[1]=1

Evaluate derivatives numerically:

In[1]:=1
✖
https://wolfram.com/xid/0pniox72hyojcfab-keoz73
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0pniox72hyojcfab-7ocy8q
Out[2]=2

RamanujanTauZ can be expressed using RamanujanTauL:

In[1]:=1
✖
https://wolfram.com/xid/0pniox72hyojcfab-xeprj7
Out[1]=1
Wolfram Research (2007), RamanujanTauL, Wolfram Language function, https://reference.wolfram.com/language/ref/RamanujanTauL.html.
✖
Wolfram Research (2007), RamanujanTauL, Wolfram Language function, https://reference.wolfram.com/language/ref/RamanujanTauL.html.

Text

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

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

CMS

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

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

APA

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

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

BibTeX

@misc{reference.wolfram_2025_ramanujantaul, author="Wolfram Research", title="{RamanujanTauL}", year="2007", howpublished="\url{https://reference.wolfram.com/language/ref/RamanujanTauL.html}", note=[Accessed: 21-April-2025 ]}

✖
@misc{reference.wolfram_2025_ramanujantaul, author="Wolfram Research", title="{RamanujanTauL}", year="2007", howpublished="\url{https://reference.wolfram.com/language/ref/RamanujanTauL.html}", note=[Accessed: 21-April-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_ramanujantaul, organization={Wolfram Research}, title={RamanujanTauL}, year={2007}, url={https://reference.wolfram.com/language/ref/RamanujanTauL.html}, note=[Accessed: 21-April-2025 ]}

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@online{reference.wolfram_2025_ramanujantaul, organization={Wolfram Research}, title={RamanujanTauL}, year={2007}, url={https://reference.wolfram.com/language/ref/RamanujanTauL.html}, note=[Accessed: 21-April-2025 ]}