Certain functional equations for the Riemann Zeta function?

2012-05-25T10:15:11Z

Referring to this question I asked on math.SE. I am posting a more generalized question here, for answers and further inquiry.

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 $\zeta(s+1)$, or $\zeta(s)$, $\zeta(s+1)$, and $\zeta(s+2)$ or in general $\zeta(s)$, $\zeta(s+1)$, $\zeta(s+2)$, ..., $\zeta(s+n)$ for $\Re(s) > 0$ and for $n \in \mathbb{N}$?

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.

My formula goes as follows: For any $n \in \mathbb{N}$ and $\Re(s) > 0$ we have $$\frac{1}{s-1} + \sum_{r=1}^n \binom{n}{r} (-1)^r \left(\frac{\zeta(s+r-1)}{s+n-1} + \int_1^\infty\frac{\rho(x)^r}{x^{s+r}}\mathrm{d}x\right) = 0$$ where $\binom{n}{r}$ is the binomial coefficient and $\rho(x)$ is the fractional part of $x$

For example, putting $n=1$ we get the well known identity, $$\frac{1}{s-1} - \frac{\zeta(s)}{s} - \int_1^\infty\frac{\rho(x)}{x^{s+1}}\mathrm{d}x = 0$$ putting $n=2$, $$\frac{1}{s-1} - 2\left(\frac{\zeta(s)}{s+1} + \int_1^\infty\frac{\rho(x)}{x^{s+1}}\mathrm{d}x\right) + \left(\frac{\zeta(s+1)}{s+1} + \int_1^\infty\frac{\rho(x)^2}{x^{s+2}}\mathrm{d}x\right) = 0$$ and so on...

EDIT: Classic answer by Juan. This question is now solved.

Answer by juan for Certain functional equations for the Riemann Zeta function?

2012-05-28T07:45:11Z

Equations of this type are known. You may see, for example, the classical book "Primzahlen" by Landau paragraph 67. "Continuation of the zeta function by partial integration"

There it is proved the formula $$ (s-1)(\zeta(s)-1)-1=-\frac{(s-1)s}{2!}(\zeta(s+1)-1)- \frac{(s-1)s(s+1)}{3!}(\zeta(s+2)-1)-\cdots$$ $$\cdots-\frac{(s-1)s\cdots(s+q)}{(q+2)!}(\zeta(s+q+1)-1) -\frac{(s-1)s\cdots(s+q+1)}{(q+2)!}\int_1^\infty \frac{\rho(x)^{q+2}}{x^{s+q+2}} dx.$$

From this we may get the beautiful formula $$1=\binom{s-1}{1}(\zeta(s)-1)+\binom{s}{2}(\zeta(s+1)-1)+\binom{s+1}{3}(\zeta(s+2)-1)+ \cdots$$ (See Titchmarsh 2.14, formula (2.14.1) and also (2.14.2))

From the formula in Landau you see that the integral $\int_1^\infty \frac{\rho(x)^r}{x^{s+r}}\,dx$ can be expressed in terms of the $\zeta(s+k)$.

Certain functional equations for the Riemann Zeta function? - MathOverflow

Certain functional equations for the Riemann Zeta function?

Answer by juan for Certain functional equations for the Riemann Zeta function?