Reference: Section 14 Zeta's scenary, "Exploring Euler's Mathematics", S.Kurokawa, Springer Japan, 2007
install_github("YasuakiHonda","euler-maclaurin-sum","master");
asdf_load_source("euler-maclaurin-sum");
assume(M>N); assume(N>0); assume(M>1);assume(s>1);
ems;
renz:ems,f(n):=n^(-s),K=4,nouns;
renz1:subst(inf,M,lhs(renz))=map(lambda([exp],factor(limit(exp,M,inf))),rhs(renz));
zeta(s)=sum(n^(-s),n,1,N)+sum(n^(-s),n,N,inf)-N^(-s);
renz2:%,renz1;
renz3:lhs(renz2)=limit(rhs(substpart(0,renz2,2,1)),N,inf);
rhs(renz3),s=0;
rhs(renz3),s=-1;
%,simpsum:true,ratsimp;
rhs(renz3),s=-2;
%,simpsum:true,ratsimp;
rhs(renz3),s=-1/2;
part(%,1);
%,N:20,nouns,numer;
zeta(-1/2),numer;