I am using the following function from Tomas Oliveira e Silva
(converted by myself into Python). See
http://trac.sagemath.org/sage_trac/ticket/8135
from math import log
#
# computation of li(x^{1/2+I t})+li(x^{1/2-I t})
#
# li(x^\rho) = x^\rho/u*(1+1/u+2!/u^2+3!/u^3+...),
# with \rho=1/2+it and u=\rho\log(x)
#
def li(x,t):
u=(1/2+I*t)*log(x); #/* log(x^{1/2+I t}) */
s0=2*x^(1/2+I*t)/u;
tol=1e-9/abs(s0); #/* we desire an error of less than 10^{-9} */
s1=s2=1;
k=1;
while(True):
if(k>100):
print "li: unable to attain the desired precision"
return
s2*=k/u;
s1+=s2;
if(abs(s2)<tol):
break;
k+=1;
return(real(s0*s1));
li(10**10,39.0)
-101.969133925197
li(10**10,39)
li: unable to attain the desired precision
This sure looks like a coercion problem, but maybe I'm just doing
something wrong.
In case you are wondering why I would want to do such a computation, I
am curious about the accuracy of the explicit formula for prime_pi if
we pretend the nontrivial zeros are different. I would like to see if
the zeros can be calculated by searching for where they should be for
the best prime_pi approximation. I would like to see if this could be
done for the twin prime counting function (for which the periodic
component is a complete mystery).
Kevin Stueve
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