ZetaZero
✖
ZetaZero
Details
- Mathematical function, suitable for both symbolic and numerical manipulation.
- For positive k, ZetaZero[k] represents the zero of on the critical line that has the k smallest positive imaginary part.
- For negative k, ZetaZero[k] represents zeros with progressively larger negative imaginary parts.
- N[ZetaZero[k]] gives a numerical approximation to the specified zero.
- ZetaZero can be evaluated to arbitrary numerical precision.
- ZetaZero automatically threads over lists.
Examples
open allclose allBasic Examples (3)Summary of the most common use cases
Scope (8)Survey of the scope of standard use cases
Numerical Evaluation (3)
https://wolfram.com/xid/0en3dhgoo-cksbl4
https://wolfram.com/xid/0en3dhgoo-whe1w
https://wolfram.com/xid/0en3dhgoo-b0wt9
https://wolfram.com/xid/0en3dhgoo-zn1q5
Evaluate efficiently at high precision:
https://wolfram.com/xid/0en3dhgoo-di5gcr
https://wolfram.com/xid/0en3dhgoo-bq2c6r
Specific Values (3)
https://wolfram.com/xid/0en3dhgoo-e3n9bq
Find the first zero of Zeta[1/2+ x] using FindRoot:
https://wolfram.com/xid/0en3dhgoo-f2hrld
https://wolfram.com/xid/0en3dhgoo-cbf1qg
ZetaZero threads elementwise over lists:
https://wolfram.com/xid/0en3dhgoo-bsnwki
Visualization (2)
Display zeros of Im[Zeta[1/2+ z]] function:
https://wolfram.com/xid/0en3dhgoo-ecj8m7
Show the first zero greater than 15:
https://wolfram.com/xid/0en3dhgoo-df2nos
Generalizations & Extensions (1)Generalized and extended use cases
Negative order is interpreted as a reflected root of the Zeta function:
https://wolfram.com/xid/0en3dhgoo-enmwx
https://wolfram.com/xid/0en3dhgoo-dumcij
https://wolfram.com/xid/0en3dhgoo-g1529z
Applications (5)Sample problems that can be solved with this function
Plot distances between successive zeros:
https://wolfram.com/xid/0en3dhgoo-fjqkly
https://wolfram.com/xid/0en3dhgoo-c97ys
Compute Gram points:
https://wolfram.com/xid/0en3dhgoo-kfiebu
Show good Gram points, where RiemannSiegelZ changes sign for consecutive points:
https://wolfram.com/xid/0en3dhgoo-cxhn9k
https://wolfram.com/xid/0en3dhgoo-vrlwg
First occurrence of Lehmer's phenomenon:
https://wolfram.com/xid/0en3dhgoo-bswnd0
Plot an approximation of the number of primes and prime powers using MangoldtLambda and ZetaZero:
https://wolfram.com/xid/0en3dhgoo-ndgzfg
https://wolfram.com/xid/0en3dhgoo-5i81mx
The more zeros used, the closer the approximation:
https://wolfram.com/xid/0en3dhgoo-yqh0yo
Properties & Relations (1)Properties of the function, and connections to other functions
https://wolfram.com/xid/0en3dhgoo-k9nae1
Possible Issues (1)Common pitfalls and unexpected behavior
Wolfram Research (2007), ZetaZero, Wolfram Language function, https://reference.wolfram.com/language/ref/ZetaZero.html.
Text
Wolfram Research (2007), ZetaZero, Wolfram Language function, https://reference.wolfram.com/language/ref/ZetaZero.html.
Wolfram Research (2007), ZetaZero, Wolfram Language function, https://reference.wolfram.com/language/ref/ZetaZero.html.
CMS
Wolfram Language. 2007. "ZetaZero." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/ZetaZero.html.
Wolfram Language. 2007. "ZetaZero." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/ZetaZero.html.
APA
Wolfram Language. (2007). ZetaZero. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/ZetaZero.html
Wolfram Language. (2007). ZetaZero. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/ZetaZero.html
BibTeX
@misc{reference.wolfram_2024_zetazero, author="Wolfram Research", title="{ZetaZero}", year="2007", howpublished="\url{https://reference.wolfram.com/language/ref/ZetaZero.html}", note=[Accessed: 10-January-2025
]}
BibLaTeX
@online{reference.wolfram_2024_zetazero, organization={Wolfram Research}, title={ZetaZero}, year={2007}, url={https://reference.wolfram.com/language/ref/ZetaZero.html}, note=[Accessed: 10-January-2025
]}