Markus Schepke's profile - activity

Thanks for that!

It shouldn't matter here, since li(x) and Li(x) only differ in an additive constant, which should also hold for the complex continuation. But if in doubt - I mean Li(x)...

Yes, I think if you evaluate Li(x) = Ei(log(x)) with the definition of the complex logarithm that Sage implements by default, the value could be considered "correct". But to be honest, I am not sure how the correct mathematical complex continuation of Li is defined. But the above expression certainly gives the expected result when working in the field of analytic number theory.

It really seems come down to which branch of the complex logarithm you choose. log_integral(x) is implemented as Ei(log(x)), so it depends on the definition of the complex logarithm. Going up the critical line, the following delivers the right values for Li(20^(1/2+s*i)):

Ei((1/2+s*i)*log(20))

Riemann's formula for the number of primes less than a given number requires the calculation of the logarithmic integral (Li(x) or li(x)) for complex values. This function is implemented in Sage as log_integral.

However, it does not seem to give the correct values for complex arguments. When I feed in -4.42464733272289 + 0.649996908475887I it returns -0.0380977804390431 + 4.49840994945387I, but -4.41940689179334 - 0.684720910130221I returns -0.0281163576275170 - 4.50165559913773I, i.e., there seems to be a discontinuity when the imaginary part turns negative.

This script shows the behaviour:

start = 0
end   = 10
steps = 100

args = [(start * (1 - s / steps) + end * (s / steps)) for s in range(steps+1)]
val1 = [20^(1/2+s*i) for s in args]
val2 = [log_integral(s) for s in val1]

for (x,y,z) in zip(args, val1, val2):
    print n(x), '\t', n(y), '\t', n(z)

Maybe the problem is that a different branch of the complex logarithm should be used. But the values of the above sequence should converge to pi * i, which doesn't seem to be the case at all.

Has anybody seen this behaviour before or any idea how to fix it?