prime_pi.plot is wrong (!)

[williamstein]

I was computed Riemann's analytic formula for pi(X), and was disturbed it wasn't converging to pi(X). It turned out that the function in Sage for a while for plotting prime_pi is buggy! For example, try this:

sage: prime_pi.plot(5,10).show(gridlines='minor',frame=True)
sage: prime_pi(8)
4

You'll see a plot that has a horizontal line at height 5 on it.

This is very bad and embarrassing!

Component: number theory

Author: William Stein

Reviewer: R. Andrew Ohana, Minh Van Nguyen

Merged: Sage 4.1.2.alpha0

Issue created by migration from https://trac.sagemath.org/ticket/6811

[williamstein]

comment:1

I've attached code to fix this bug. It does things right.

(1) I created a plot_step_function command that can be used to plot general step functions, and added it to the reference manual.

(2) I changed the current broken prime_pi.plot to use that and use a much easier to understand (hence right) algorithm to generate the plot.

(3) I fixed a bunch of ReST mistakes in prime_pi.pyx while I was at it.

(4) I added prime_pi to the reference manual.

[williamstein]

Attachment: trac_6811.patch.gz

[ohanar]

comment:2

Looks fine:

1. Importing sage.plot.all is no longer necessary in prime_pi.pyx
2. May want to change

for i in range(len(v)):
    w.append(v[i])
    if i+1 < len(v):
        w.append((v[i+1][0],v[i][1]))

to

for i in range(len(v)-1):
    w.append(v[i])
    w.append((v[i+1][0],v[i][1]))
w.append(v[len(v)-1])

for readability.

1. The plot_step_function always starts horizontal and ends vertically, this can sometimes lead to rather odd looking results in my opinion. For example, compare

sage: plot_step_function([(i,i^3) for i in range(6)])
sage: plot_step_function([(i,i^3) for i in range(6)]) + line([(5,125),(6,125)])

a. If we are to make any changes to this, we would need to consider uneven intervals of definition (say the function [(i<sup>2,i</sup>3) for i in range(6)]).

1. Might be useful to use the plot_step_function elsewhere. For example, with Riemann sums it is either difficult or impossible to enable vertical lines, and the floor function is in the opposite situation.

[ohanar]

comment:3

Also, we need to fix the credit situation in prime_pi.pyx

[sagetrac-mvngu]

reviewer patch; fixes typos

[sagetrac-mvngu]

comment:4

Attachment: trac_6811-reviewer.patch.gz

The reviewer patch trac_6811-reviewer.patch fixes some typos in trac_6811.patch. One of these typos results in a warning when building the reference manual.

[sagetrac-mvngu]

comment:5

Merged both patches.

[sagetrac-mvngu]

Author: William Stein

[sagetrac-mvngu]

Merged: Sage 4.1.2.alpha0

[sagetrac-mvngu]

Reviewer: R. Andrew Ohana, Minh Van Nguyen