Certain functional equations for the Riemann Zeta function? - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T18:18:34Z http://mathoverflow.net/feeds/question/97929 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/97929/certain-functional-equations-for-the-riemann-zeta-function Certain functional equations for the Riemann Zeta function? Roupam Ghosh 2012-05-25T10:15:11Z 2012-05-28T15:34:09Z <p>Referring to <a href="http://math.stackexchange.com/questions/147377/other-functional-equations-for-zetas" rel="nofollow">this</a> question I asked on math.SE. I am posting a more generalized question here, for answers and further inquiry.</p> <blockquote> <p>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 $n \in \mathbb{N}$?</p> </blockquote> <p><em>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.</em></p> <p><em>My formula goes as follows: For any $n \in \mathbb{N}$ and $\Re(s) > 0$ we have</em> $$\frac{1}{s-1} + \sum_{r=1}^n \binom{n}{r} (-1)^r \left(\frac{\zeta(s+r-1)}{s+n-1} + \int_1^\infty\frac{\rho(x)^r}{x^{s+r}}\mathrm{d}x\right) = 0$$ <em>where $\binom{n}{r}$ is the binomial coefficient and $\rho(x)$ is the fractional part of $x$</em> </p> <p><em>For example, putting $n=1$ we get the well known identity,</em> $$\frac{1}{s-1} - \frac{\zeta(s)}{s} - \int_1^\infty\frac{\rho(x)}{x^{s+1}}\mathrm{d}x = 0$$ <em>putting $n=2$,</em> $$\frac{1}{s-1} - 2\left(\frac{\zeta(s)}{s+1} + \int_1^\infty\frac{\rho(x)}{x^{s+1}}\mathrm{d}x\right) + \left(\frac{\zeta(s+1)}{s+1} + \int_1^\infty\frac{\rho(x)^2}{x^{s+2}}\mathrm{d}x\right) = 0$$ <em>and so on...</em></p> <p>EDIT: Classic answer by Juan. This question is now solved.</p> http://mathoverflow.net/questions/97929/certain-functional-equations-for-the-riemann-zeta-function/98177#98177 Answer by juan for Certain functional equations for the Riemann Zeta function? juan 2012-05-28T07:45:11Z 2012-05-28T14:51:38Z <p>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"</p> <p>There it is proved the formula $$(s-1)(\zeta(s)-1)-1=-\frac{(s-1)s}{2!}(\zeta(s+1)-1)- \frac{(s-1)s(s+1)}{3!}(\zeta(s+2)-1)-\cdots$$ $$\cdots-\frac{(s-1)s\cdots(s+q)}{(q+2)!}(\zeta(s+q+1)-1) -\frac{(s-1)s\cdots(s+q+1)}{(q+2)!}\int_1^\infty \frac{\rho(x)^{q+2}}{x^{s+q+2}} dx.$$</p> <p>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))</p> <p>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 $\zeta(s+k)$. </p>