- #1

saltydog

Science Advisor

Homework Helper

- 1,582

- 2

## Main Question or Discussion Point

Can anyone tell me if Riemann's Prime Counting function can be solved by residue integration?

Here it is:

[tex]J(x)=\frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty}\frac{ln(\zeta(s))x^s}{s}ds[/tex]

which has the solution:

[tex]J(x)=li(x)-\sum_{\rho}li(x^\rho)-ln(2)+

\int_x^{\infty}\frac{dt}{t(t^2-1)ln(t)}[/tex]

I mean it has four sets of poles: 0,1, trivial zeros, complex zeros (due to log term) and I would suspect each of the four terms above may result from the residues there assuming the integral goes to zero around the remainder of a closed contour.

Anyway I'll be looking into it as well.

Here it is:

[tex]J(x)=\frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty}\frac{ln(\zeta(s))x^s}{s}ds[/tex]

which has the solution:

[tex]J(x)=li(x)-\sum_{\rho}li(x^\rho)-ln(2)+

\int_x^{\infty}\frac{dt}{t(t^2-1)ln(t)}[/tex]

I mean it has four sets of poles: 0,1, trivial zeros, complex zeros (due to log term) and I would suspect each of the four terms above may result from the residues there assuming the integral goes to zero around the remainder of a closed contour.

Anyway I'll be looking into it as well.

Last edited: