4
$\begingroup$

The Riemann zeta function, initially defined as $$\zeta(s)=\sum_{n=1}^\infty \frac{1}{n^s} $$ for $\Re(s)>1$ also has infinite sum representations on larger domains. One such sum is $$\zeta(s)=\frac{1}{s-1}\sum_{n=1}^\infty \left(\frac{n}{(n+1)^s}-\frac{n-s}{n^s}\right), $$ defined for $\Re(s)>0$. Moreover, for any nonnegative integer $k$ there is a sum of this nature, analytic the half-plane $\Re(s) >-k$. I would like to know whether the Euler product representaiton $$\zeta(s)=\prod_{p \text{ prime}} \frac{1}{1-p^{-s}}, \; \Re(s)>1 $$ also has some modified forms which converge on larger domains.

$\endgroup$
5
$\begingroup$

arXiv:1208.1440 has this generalized Euler product on $\Re(s)>0$:

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.