zetafunction
Jun1-09, 05:53 AM
in a seminar at my univesity we had a brief introduction to prime number theorem , in the blackboard the professor wrote
\psi(x) =x-\sum_\rho\frac{x^\rho}{\rho} - \log(2\pi) -\log(1-x^{-2})/2
and he also gave an introduction (gave no proofs since we were in a seminar an were in a hurry) to the thing he called L-function
L(\chi,s)}=\sum_{n=1}^{\infty} \frac{\chi(n)}{n^s}
and he said that 'Riemann zeta can be obtained from this ,
My question is , can we have an explicit formula in the form
? =x-\sum_\rho\frac{x^\rho}{\rho} - G(x)
but with the 'zeros' of the L-fucntions ? , i mean we should evaluate the contour integral
2i \pi f(x)= \int_{c-i\infty}^{c+i\infty}ds \frac{ L'(\chi,s)}{ L(\chi,s)}\frac{x^{s}}{s}
\psi(x) =x-\sum_\rho\frac{x^\rho}{\rho} - \log(2\pi) -\log(1-x^{-2})/2
and he also gave an introduction (gave no proofs since we were in a seminar an were in a hurry) to the thing he called L-function
L(\chi,s)}=\sum_{n=1}^{\infty} \frac{\chi(n)}{n^s}
and he said that 'Riemann zeta can be obtained from this ,
My question is , can we have an explicit formula in the form
? =x-\sum_\rho\frac{x^\rho}{\rho} - G(x)
but with the 'zeros' of the L-fucntions ? , i mean we should evaluate the contour integral
2i \pi f(x)= \int_{c-i\infty}^{c+i\infty}ds \frac{ L'(\chi,s)}{ L(\chi,s)}\frac{x^{s}}{s}