4
$\begingroup$

For the Riemann zeta function, we know of the standard functional equation that relates $\zeta(s)$ and $\zeta(1-s)$. I wanted to know whether there are functional equations that relates $\zeta(s)$ and $\zeta(s-1)$?

EDIT: My main motivation behind asking this question is I have found such an equation, but I do not know whether such an equation exists in literature. Also, I do not want to appear as if I am promoting my formula here, but rather I am more interested in the works that have been done in such directions.

As per @lhf's request here is my formula, for $\Re(s) > 1$ $$ \zeta(s) + \frac{2}{s-1}\zeta(s-1) = \frac{s}{s-2} - s\int_1^\infty \frac{\{x\}^2}{x^{s+1}} dx$$ where $\{x\}$ is the fractional part of x.

  • 3
    In the conference volume *Algebraic Number Fields: L-functions and Galois Properties* (1977), there is an article by Audrey Terras with the title "A relation between $\zeta_K(s)$ and $\zeta_K(s-1)$ for any algebraic number field $K$".2012-05-20
  • 1
    Just show us your equation and ask whether it is known. It is probably a simpler question to answer than the current one.2012-05-20
  • 0
    @lhf "I am more interested in the works that have been done in such directions" (OK I will post it, not a big deal, except the part that it might be incorrect. So, please let me check that before I post it!).2012-05-20
  • 0
    @KCd Thanks I will check that out.2012-05-20

3 Answers 3