--- In

primenumbers@yahoogroups.com,

"Cino Hilliard" <hillcino368@...> wrote:

> I want to examine

> sm(m,n)=sum(x=m,n,x/log(x)-1/log(x)+a/log(x)^2-b/log(x)^3)

> a=1,b=1,m=2

f(x,lx) = x/lx - 1/lx + a/lx^2 - b/lx^3; \\ modify at will

a=1; b=1; m=2; \\ chosen values

\p50

terms=10;trunc=10^4;g=f(x,lx);d=[];bv=bernvec(terms+1);

{for(n=1,2*terms-1, \\ store odd derivatives

g=deriv(g,x)+1/x*deriv(g,lx);if(n%2,d=concat(d,g)));}

fx(x)=f(x,log(x));

dx(y)=subst(subst(d,x,y),lx,log(y));

{emac(m,n)=local(dm,dn);

dm=dx(m);dn=dx(n);intnum(x=m,n,fx(x))+(fx(n)+fx(m))/2

+sum(k=1,terms,bv[k+1]/((2*k)!)*(dn[k]-dm[k]));}

s=sum(k=2,trunc-1,fx(k));

{sm(m,n) = \\ as requested

if(n>trunc&&trunc>m&&m>1,s-sum(k=2,m-1,fx(k))+emac(trunc,n));}

for(k=10,22,print([k,sm(m,sqrtint(10^k))]))

realprecision = 57 significant digits (50 digits displayed)

[10, 455051173.78013348859945302791104662093667330309988]

[11, 4118034570.4638268343263084222227560024582402009128]

[12, 37607913583.517014991610186636613681414323919745603]

[13, 346065399465.03248684994266161913298076743032790216]

[14, 3204941752475.7266213978929589359007763992788467339]

[15, 29844569450368.625194208305114011697818820369920647]

[16, 279238341514801.87986838501363369093160129071417051]

[17, 2623557157209703.2566630532132281009952796686474077]

[18, 24739954285430064.167128134429584525396645626800130]

[19, 234057667279245072.15948934429135642246573100181670]

[20, 2220819602565566257.9962162533153260118661266449421]

[21, 21127269485065195786.472528704950924409440892097567]

[22, 201467286689267167012.50527750985870920146256019519]

Comment: This is an inefficient method of fiddling with Li(x).

Cino is trying to avoid integration. Yet it is needed for

Euler-Maclaurin summation.

David