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## numtheory package

Maple 15

Greetings to all.

I have been using the numtheory package for quite some time now and it has helped me advance on a number of problems. Recently an issue came to my attention that I have known about for a long time but somehow never realized that it can be fixed. This is the fact that the numtheory package does not know about Dirichlet series, finite and infinite. Here are two links: this one, where I calculate the dominant pole of an infinite Dirichlet series using Maple and this one, which does not include Maple code, where I calculate three classic infinite Dirichlet series. There are four main issues here:

• there is no data type for finite Dirichlet series that numtheory knows about
• the sum command does not recognize Dirichlet series, e.g.
`sum(tau(n)/n^s, n=1..infinity);`
will fail
• there is no coefficient extraction operator for Dirichlet series
• Dirichlet convolutions and Dirichlet inverses of multiplicative functions are not implemented

Here are some comments on these issues.

• First, it appears to me imperative that Maple provide support for Dirichlet series. This is part of the undergraduate curriculum and we have here a deficit that should have no place in a product like Maple.
• Second, a naive user might well ask what the purpose is of providing all those numbertheoretic functions (which are extremely useful btw) if the sum command does not know about them. This will force anybody working with serious Mellin transforms in number theory to revert to pen and paper, when in fact this should be an area where Maple ought to shine. Unfortunately summing those Dirichlet series requires sophisticated algorithms and possibly some table lookup as a last resort, like in the catalog by Gould and Shonhiwa, where about eighty such series are listed. Others that are rational functions of finite Dirichlet series can be calculated algorithmically, as shown in the Maple code at the first link.
• Third, there being no coefficient extraction I had to program one myself to deal with cases where simplify turns Dirichlet series terms into the wrong format, absorbing the coefficent on the term into the complex exponential. (This code too can be seen at the first link.) Coefficient extractions of standard series can be done by table lookup and algorithmically for rational functions of finite Dirichlet series. Coefficient extraction could be immensely powerful on classic Dirichlet series, returning a closed form.
• Fourth, it is extremely useful when one is confronted with a new numbertheoretic function, to study its Dirichlet inverse and its convolutions with other basic numbertheoretic functions in order to discover patterns and laws. I do this all the time. I had to write the code for inverses and convolutions myself (it is only a few lines). This should be part of the numtheory package. Moreover the Dirichlet series corresponding to a convolution can be summed in closed form as the product of the Dirichlet series for the two functions in the convolution.

I hope these points will perhaps initiate a discussion and help improve Maple. Some of what I have mentioned is very simple, and some of the other proposals require sophisticated algorithms. Not being a Maple wizard I might have overlooked some tricks that accomplish some of the tasks listed above.

Best regards, Marko Riedel ﻿