keepfloat:true$
assume(x>1)$
texput(moebius,"\\mu")$
declare(rho,complex)$

J(x)=li(x)-sum(li(x^rho[i])+li(x^(conjugate(rho[i]))),i,1,inf)-log(2)+integrate(1/(t*(t^2-1)*log(t)),t,x,inf);

pi(x)=sum(moebius(m)/m*J(x^(1/m)),m,1,floor(log(x)/log(2)));

pi1[n]:=if n<2 then 0 elseif primep(n) then pi1[n-1]+1 else pi1[n-1]$

pi(x):=if integerp(x) and primep(x) then pi1[x]-1/2 else pi1[floor(x)]$

plot2d(pi(x),[x,0,100])$

J2(x):=expintegral_li(x)-log(2)+romberg(1/(t*(t^2-1)*log(t)),t,x,100);

pi2(x):=ev(sum(moebius(m)/m*J2(x^(1/m)),m,1,floor(log(x)/log(2))),numer);

plot2d([pi(x),pi2(x)],[x,2,1000])$

expintegral_ei((1/2+%i*t)*log(x))=expintegral_li(x^(1/2+%i*t));

draw2d(nticks=1000,
       parametric( realpart(expintegral_ei(expand((1/2+%i*t)*log(20.0)))),
                   imagpart(expintegral_ei(expand((1/2+%i*t)*log(20.0)))),
                   t,-50,50) ,
       color=red,
       parametric( realpart(expintegral_li(rectform(20.0^(1/2+%i*t)))),
                   imagpart(expintegral_li(rectform(20.0^(1/2+%i*t)))),
                   t,0,2.1));

Li_power(x,t):=2*realpart(expintegral_ei(expand((1/2+%i*t)*log(x))));

/*
The LMFDB Collaboration, The L-functions and Modular Forms Database, 
home page of the Zeros of zeta(s),
https://www.lmfdb.org/zeros/zeta/?limit=200&N=1, 2020 , [Online; accessed 10 October 2020].
*/
img_rho:[
14.1347251417346937904572519835625,
21.0220396387715549926284795938969,
25.0108575801456887632137909925628,
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393.4277438444340259366989529201288,
395.5828700109937209708777113231417,
396.3818542225921869319994544917305
]$

J3(x):=
expintegral_li(x)
-sum(Li_power(x,img_rho[i]),i,1,200)
-log(2.0)
+romberg(1/(t*(t^2-1)*log(t)),t,x,20);

pi3(x):=ev(sum(moebius(m)/m*J3(x^(1/m)),m,1,floor(log(x)/log(2))),numer);

plot2d([pi3(x),pi(x)],[x,3,200]);