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RamanujanTauTheta
Details
- Mathematical function, suitable for both symbolic and numerical manipulation.
- for real .
- arises in the study of the Ramanujan L-function on the critical line. It is closely related to the number of zeros of for .
- Apart from a sign, gives the phase of the Ramanujan L-function .
- is an analytic function of except for branch cuts on the imaginary axis running from to .
- For certain special arguments, RamanujanTauTheta automatically evaluates to exact values.
- RamanujanTauTheta can be evaluated to arbitrary numerical precision.
- RamanujanTauTheta automatically threads over lists.
Examples
open allclose allBasic Examples (6)Summary of the most common use cases
https://wolfram.com/xid/0mfepfe68sn7ny4c2e-slcg8i
Plot over a subset of the reals:
https://wolfram.com/xid/0mfepfe68sn7ny4c2e-dx9l95
Plot over a subset of the complexes:
https://wolfram.com/xid/0mfepfe68sn7ny4c2e-kiedlx
Series expansion at the origin:
https://wolfram.com/xid/0mfepfe68sn7ny4c2e-fdkkja
Series expansion at Infinity:
https://wolfram.com/xid/0mfepfe68sn7ny4c2e-20imb
Series expansion at a singular point:
https://wolfram.com/xid/0mfepfe68sn7ny4c2e-d2klx1
Scope (27)Survey of the scope of standard use cases
Numerical Evaluation (7)
https://wolfram.com/xid/0mfepfe68sn7ny4c2e-l274ju
https://wolfram.com/xid/0mfepfe68sn7ny4c2e-pmo0yc
https://wolfram.com/xid/0mfepfe68sn7ny4c2e-b0wt9
https://wolfram.com/xid/0mfepfe68sn7ny4c2e-zn1q5
The precision of the output tracks the precision of the input:
https://wolfram.com/xid/0mfepfe68sn7ny4c2e-y7k4a
https://wolfram.com/xid/0mfepfe68sn7ny4c2e-hfml09
https://wolfram.com/xid/0mfepfe68sn7ny4c2e-dm5qi7
Evaluate efficiently at high precision:
https://wolfram.com/xid/0mfepfe68sn7ny4c2e-di5gcr
https://wolfram.com/xid/0mfepfe68sn7ny4c2e-bq2c6r
Compute average-case statistical intervals using Around:
https://wolfram.com/xid/0mfepfe68sn7ny4c2e-cw18bq
Compute the elementwise values of an array:
https://wolfram.com/xid/0mfepfe68sn7ny4c2e-thgd2
Or compute the matrix RamanujanTauTheta function using MatrixFunction:
https://wolfram.com/xid/0mfepfe68sn7ny4c2e-o5jpo
Specific Values (2)
https://wolfram.com/xid/0mfepfe68sn7ny4c2e-cgkwdk
Find positive minimum of RamanujanTauTheta[x]:
https://wolfram.com/xid/0mfepfe68sn7ny4c2e-f2hrld
https://wolfram.com/xid/0mfepfe68sn7ny4c2e-kt8qt6
Visualization (2)
Plot the RamanujanTauTheta:
https://wolfram.com/xid/0mfepfe68sn7ny4c2e-b1j98m
Plot the real part of RamanujanTauTheta function:
https://wolfram.com/xid/0mfepfe68sn7ny4c2e-8gdas
Plot the imaginary part of RamanujanTauTheta function:
https://wolfram.com/xid/0mfepfe68sn7ny4c2e-eweler
Function Properties (10)
RamanujanTauTheta is defined for all real values:
https://wolfram.com/xid/0mfepfe68sn7ny4c2e-cl7ele
https://wolfram.com/xid/0mfepfe68sn7ny4c2e-de3irc
Function range of RamanujanTauTheta:
https://wolfram.com/xid/0mfepfe68sn7ny4c2e-evf2yr
RamanujanTauTheta threads over lists:
https://wolfram.com/xid/0mfepfe68sn7ny4c2e-ikm0le
RamanujanTauTheta is an analytic function of x:
https://wolfram.com/xid/0mfepfe68sn7ny4c2e-h5x4l2
RamanujanTauTheta is neither non-increasing nor non-decreasing:
https://wolfram.com/xid/0mfepfe68sn7ny4c2e-g6kynf
RamanujanTauTheta is not injective:
https://wolfram.com/xid/0mfepfe68sn7ny4c2e-gi38d7
https://wolfram.com/xid/0mfepfe68sn7ny4c2e-ctca0g
RamanujanTauTheta is surjective:
https://wolfram.com/xid/0mfepfe68sn7ny4c2e-hkqec4
https://wolfram.com/xid/0mfepfe68sn7ny4c2e-hdm869
RamanujanTauTheta is neither non-negative nor non-positive:
https://wolfram.com/xid/0mfepfe68sn7ny4c2e-84dui
RamanujanTauTheta has no singularities or discontinuities:
https://wolfram.com/xid/0mfepfe68sn7ny4c2e-mdtl3h
https://wolfram.com/xid/0mfepfe68sn7ny4c2e-mn5jws
RamanujanTauTheta is neither convex nor concave:
https://wolfram.com/xid/0mfepfe68sn7ny4c2e-kdss3
Differentiation (2)
First derivative with respect to :
https://wolfram.com/xid/0mfepfe68sn7ny4c2e-krpoah
Higher derivatives with respect to :
https://wolfram.com/xid/0mfepfe68sn7ny4c2e-z33jv
Plot the higher derivatives with respect to :
https://wolfram.com/xid/0mfepfe68sn7ny4c2e-gkzteu
Series Expansions (4)
Find the Taylor expansion using Series:
https://wolfram.com/xid/0mfepfe68sn7ny4c2e-ewr1h8
Plots of the first three approximations around :
https://wolfram.com/xid/0mfepfe68sn7ny4c2e-binhar
Find the series expansion at Infinity:
https://wolfram.com/xid/0mfepfe68sn7ny4c2e-syq
Find the series expansion for an arbitrary symbolic direction :
https://wolfram.com/xid/0mfepfe68sn7ny4c2e-t5t
Taylor expansion at a generic point:
https://wolfram.com/xid/0mfepfe68sn7ny4c2e-jwxla7
Applications (4)Sample problems that can be solved with this function
Contour plot of the absolute value of RamanujanTauTheta:
https://wolfram.com/xid/0mfepfe68sn7ny4c2e-5glvtj
The first 10 Gram points of RamanujanTauL:
https://wolfram.com/xid/0mfepfe68sn7ny4c2e-jtr6kv
Plot of RamanujanTauZ and Gram points:
https://wolfram.com/xid/0mfepfe68sn7ny4c2e-5ai4n1
Show interlacing of the roots of Sin[RamanujanTauTheta[t] and RamanujanTauZ[t]:
https://wolfram.com/xid/0mfepfe68sn7ny4c2e-ewi0mv
The number of zeros on the critical strip from 0 to :
https://wolfram.com/xid/0mfepfe68sn7ny4c2e-j7pggl
https://wolfram.com/xid/0mfepfe68sn7ny4c2e-8reysz
Properties & Relations (3)Properties of the function, and connections to other functions
RamanujanTauTheta is related to LogGamma:
https://wolfram.com/xid/0mfepfe68sn7ny4c2e-d0afjz
https://wolfram.com/xid/0mfepfe68sn7ny4c2e-fo61ya
On the critical line, RamanujanTauTheta gives the phase of RamanujanTauL up to a sign:
https://wolfram.com/xid/0mfepfe68sn7ny4c2e-fljiqq
https://wolfram.com/xid/0mfepfe68sn7ny4c2e-4gabay
RamanujanTauZ can be expressed in terms of RamanujanTauTheta and RamanujanTauL:
https://wolfram.com/xid/0mfepfe68sn7ny4c2e-imsrup
https://wolfram.com/xid/0mfepfe68sn7ny4c2e-9pbwjn
Possible Issues (1)Common pitfalls and unexpected behavior
Neat Examples (2)Surprising or curious use cases
https://wolfram.com/xid/0mfepfe68sn7ny4c2e-rirfxt
Riemann surface of RamanujanTauTheta:
https://wolfram.com/xid/0mfepfe68sn7ny4c2e-lwrstt
Wolfram Research (2007), RamanujanTauTheta, Wolfram Language function, https://reference.wolfram.com/language/ref/RamanujanTauTheta.html.
Text
Wolfram Research (2007), RamanujanTauTheta, Wolfram Language function, https://reference.wolfram.com/language/ref/RamanujanTauTheta.html.
Wolfram Research (2007), RamanujanTauTheta, Wolfram Language function, https://reference.wolfram.com/language/ref/RamanujanTauTheta.html.
CMS
Wolfram Language. 2007. "RamanujanTauTheta." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/RamanujanTauTheta.html.
Wolfram Language. 2007. "RamanujanTauTheta." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/RamanujanTauTheta.html.
APA
Wolfram Language. (2007). RamanujanTauTheta. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/RamanujanTauTheta.html
Wolfram Language. (2007). RamanujanTauTheta. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/RamanujanTauTheta.html
BibTeX
@misc{reference.wolfram_2024_ramanujantautheta, author="Wolfram Research", title="{RamanujanTauTheta}", year="2007", howpublished="\url{https://reference.wolfram.com/language/ref/RamanujanTauTheta.html}", note=[Accessed: 08-January-2025
]}
BibLaTeX
@online{reference.wolfram_2024_ramanujantautheta, organization={Wolfram Research}, title={RamanujanTauTheta}, year={2007}, url={https://reference.wolfram.com/language/ref/RamanujanTauTheta.html}, note=[Accessed: 08-January-2025
]}