Generating function of Mangoldt function

From the Mangoldt function definition with dsum(), it is easy to obtain the generating function of Mangoldt function.

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 contains the definition of dsum().
(%i1) load("elementaryNumberTheory.mac");


(%i2) mangoldt(N):=if integerp(N)
then apply(dsum,[moebius(N/d)*log(d),d,N])
else dsum(moebius(N/d)*log(d),d,N)$

(%i3) texput(nounify(mangoldt), "\\Lambda")$

 
The definition of Mangoldt function is given above.
(%i4) 'mangoldt(n)=mangoldt(n);

 

We can see that the definition works fine.

(%i5) 'mangoldt(27)=mangoldt(27);


(%i6) 'mangoldt(27)=radcan(mangoldt(27));


(%i7) 'mangoldt(28)=mangoldt(28);


(%i8) 'mangoldt(28)=radcan(mangoldt(28));


 

The following holds for any number-theoretic functions:

(%i9) dprod;

 

Having a(n) being Moebius function and b(n) being logarithmic function, the following holds:
(%i10) dprod,a(n):=moebius(n),b(n):=log(n);


 

Since (%o11) and (%o12) holds, and right hand side of (%o10) is already the generating function of Mangoldt function, we obtain (%o13).

(%i11) 'diff(zeta(s),s)=sum(diff(1/n^s,s),n,1,inf);


(%i12) 1/zeta(s)=sum(moebius(n)/n^s,n,1,inf);


(%i13) -diff(zeta(s),s)/zeta(s)=sum('mangoldt(n)/n^s,n,1,inf);


(%i14)

 

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)