Make symbolic versions of moebius, sigma, and euler_phi functions

[zimmermann6]

I hit the following problem:

sage: f(x) = sigma(x)-x
...
TypeError: unable to convert x (=x) to an integer

Wouldn't it better to keep sigma(x) unevaluated for x not an integer?
Note that f = lambda(x):sigma(x)-x works but it less nice.

CC: @nexttime @benjaminfjones @eviatarbach @slel

Component: symbolics

Issue created by migration from https://trac.sagemath.org/ticket/8383

[kcrisman]

comment:1

We'd have to produce a symbolic version of sigma somewhere. Do you want this for all arithmetic functions? We really should have an arithmetic function class anyway, but it's probably more trouble than it's worth to actually do it.

[zimmermann6]

comment:2

Replying to @kcrisman:

We'd have to produce a symbolic version of sigma somewhere. Do you want this for all arithmetic functions?

yes, in Maple for example numtheorysigma remains symbolic, and does not produce an error.

[kcrisman]

comment:3

Okay, then I think I will update the summary of this. Also changing component since it's more at symbolics than calculus.

We would need to have a uniform error message as well, and hopefully use plot_step_function as a unified plotting method (?).

[burcin]

comment:4

Can you either provide a list of "arithmetic functions" which should be made symbolic, or just make this ticket about sigma()?

Tickets with blanket statements about symbolic functions (see #4102, #1158, #4229) are hard to attack since nobody takes on the task of making a list of functions which need to be fixed.

[kcrisman]

comment:5

Well, at the very least the ones in rings/arith.py which have 'standard' representations should be, so Moebius, Euler_Phi, Sigma. Someday we will hopefully also implement things like the Mertens function (not to be confused with the constant), and those should also be able to remain symbolic. If Paul has others which we have and Maple leaves symbolic, that would be great.

[zimmermann6]

comment:6

Replying to @burcin:

Can you either provide a list of "arithmetic functions" which should be made symbolic, or just make this ticket about sigma()?

doesn't Sage provide such a list? It would then be easy to do a loop over all functions and
determine those which don't work with symbolic arguments.

Paul

[kcrisman]

comment:7

But I don't think we want ALL such functions (if you are referring to all functions in rings/arith.py). I don't think we have a keyword otherwise, and it certainly isn't worth the time to do four_squares or primitive_root (which just gives the smallest one) as a symbolic function before we have even implemented some of these other functions. Anyway, I'll change the summary again to make my preference clear :)

Interestingly, these three functions all give different errors upon giving them 'x' as an argument. That's probably irrelevant, but still fun to point out.