Mantoldt function

This article defines the Mangoldt function.  The file of this session can be found here. 

Maxima 5.20.0 http://maxima.sourceforge.net
using Lisp CMU Common Lisp 19f (19F)
Distributed under the GNU Public License. See the file COPYING.
Dedicated to the memory of William Schelter.
The function bug_report() provides bug reporting information.
(%i1) block(load(("/Users/yasube/Programming/imaxima-imath/imaxima.lisp")), linenum:0)$

The following file defines the function dsum(). You can find the definition of dsum() in the previous article.

(%i1) load("elementaryNumberTheory.mac");


(%i2) matchdeclare([a,b],true)$

(%i3) defrule(logexpon, a*log(b), log(b))$
The above rule logexpon defines a pattern matching and transformation function that appears in the standard definition of the Mangoldt function, which is as follows:

(%i4) mangoldt1(N):= if integerp(N)
then block([exp], if (exp:logexpon(radcan(log(N)))) = false then 0 else exp)
else apply(nounify(mangoldt1), [N])$

The above defines that if N is power of a prime, then the log of the prime is returned, else 0 is returned.

(%i5) mangoldt(N):=if integerp(N)
then apply(dsum,[moebius(N/d)*log(d),d,N])
else dsum(moebius(N/d)*log(d),d,N)$
The above defines that mangoldt function of N is defined to be sum of convolution multiplication of moebius() and log() over the divisors of N.
(%i6) texput(nounify(mangoldt), "\\Lambda")$

(%i7) a:radcan(log(11^3));


(%i8) logexpon(a);


(%i9) 'mangoldt1(11^3)=mangoldt1(11^3);

The above shows that the mangoldt1() works fine if the number is power of a prime.

(%i10) a:radcan(log(2^2*3^3));


(%i11) logexpon(a);


(%i12) 'mangoldt1(2^2*3^3)=mangoldt1(2^2*3^3);


The above shows that mangoldt1() works fine for numbers other than the power of prime.

The following session shows that the definition based on dsum() function works equally.

(%i13) a:11^3;


(%i14) 'mangoldt(a)=mangoldt(a);


(%i15) radcan(%);


(%i16) a:2^2*3^3;


(%i17) 'mangoldt(a)=mangoldt(a);


(%i18) radcan(%);


(%i19)