Solutions to differential delay equations

[Sary Drappeau]

Hi,

Sage currently has a nice implementation of the Dickman rho function which is a solution to a specific differential delay equation (DDE).

In the context of ongoing work with integrals from sieve theory, I'm considering proposing two minor changes :

1. allowing dickman_rho to return a rigorously proven RIF value (easy). As it stands, it seems the precision is a constant multiple of abs_prec for small values of the argument (for large values of the argument using saddle-point method, the relative precision is O(1/x) but the constant is non-explicit as far as I know, I don't intend to work this out)
2. implementing the Buchstab function, which is another solution to a differential delay equation, or more general solutions to DDE of the type  x f'(x) + a f(x) + b f(x-1) = 0 using the same method as in dickman_rho (Marsaglia-Zaman-Marsaglia).

Regarding 2. I'm not sure where this should belong: by default I'll propose a builtin "number theoretic function" with parameters a, b, in the same file as dickman_rho, but perhaps it should be part of a specific class, "DDESolution" ?

===

Sary

[Dima Pasechnik]

Just to remark that you are referring to Sage 9.4 docs, rather than to
the up to date ones, which should be

https://doc.sagemath.org/html/en/reference/functions/sage/functions/transcendental.html#sage.functions.transcendental.DickmanRho

(probably no difference here, but just in case)

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