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Definition of GramPoint[ ] in MathWorld's Gram Point and Gram’s Law?

Posted 15 days ago
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In the historical calculations of the Riemann zeta function Gram points and Gram’s Law plays a part. Gram’s Law, which does not always hold, says that the zeros of the Riemann zeta function, actually Z(t), are separated by the Gram points. This observation is useful to help count the number of zeros on the critical line.

Wolfram MathWorld has an article on each of Gram Point and Gram’s Law. Both have a Mathematica notebook for download.

In the Gram Point notebook there is a plot of Gram Points made with

ListPlot[Table[{n, GramPoint[n]}, {n, 0, 20}], 
 PlotStyle -> {PointSize[.02], Red}, 
 AxesLabel -> TraditionalForm /@ {n, Subscript[g, n]}]

Note the function GramPoint[n]. Since the notebook doesn’t define it, I inferred it is a built in function. However, it is not included in Mathematic 13.1.

Can somebody help me find the built in function? Had it been in an earlier version and removed?

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POSTED BY: Dan O'Leary
5 Replies
Posted 15 days ago

Hi Dan,

In the documentation for ZetaZero there is a definition for the Gram point function

gp[k_] := 
 Block[{t}, 
  t /. FindRoot[RiemannSiegelTheta[t] - Pi k, {t, (1/4 + 2 k) Pi/ProductLog[(1/8 + k)/E]}]]

Using that definition, the results in GramPoint.nb and GramsLaw.nb can be reproduced.

POSTED BY: Rohit Namjoshi

Rohit, Thank you for pointing this out. I’m looking forward to trying it.

I infer that the function GramPoint[] used in the notebooks is not a Mathematica built-in function.

Regards, Dan

POSTED BY: Dan O'Leary

I have a few more questions about this series of notebooks. In addition to the function GramPoint[] they use << UtilitiesPlot and PlotRoots[], neither of which are Mathematica 13.1.

PlotRoots[] seems to have some nice features. I could not find it in the documentation.

Is there a version available in Mathematica 13.1?

POSTED BY: Dan O'Leary

Hi Dan,

Your question led me on a quest to find the Utilities`Plot package. I found this post on MSE and downloaded the package from the Wayback Machine (attached Plot.m). Unfortunately, many notebooks and WL code on MathWorld were written for older versions of WL and have not been ported to work with more recent versions. The package will not load in V13.1.

I adapted the code for PlotRoots to work in V13.1 (probably any version after 11). See attached PlotRoots.nb.

Attachments:
POSTED BY: Rohit Namjoshi

Rohit,

Thank you very much! The notebook is very nice.

Outside of Mr. Peabody and his boy Sherman I didn’t know there was a Wayback Machine.

It would be great if the Math World code were updated so it could run on the newer versions of Mathematica.

Regards, Dan

POSTED BY: Dan O'Leary
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