This is actually not very interesting. Zeta Functions and Polylogarithms Zeta: Specific values. However, its values at odd positive integers remains mysterious to this day. \end{align} By using our site, you acknowledge that you have read and understand our Cookie Policy, Privacy Policy, and our Terms of Service. \cot z = \frac{1}{z} - 2 \sum_{k=1}^\infty \frac{z}{k^2 \pi^2 -z^2} \begin{align} What is known about the pattern for \zeta(2n+1)? [1] Bruce C. Berndt, Elementary Evaluation of ζ(2n), Math. The function \pi\tan(\pi z) has a residue -1 at z=n-\frac12 for n\in\mathbb{Z}. Proof. Another good source of information which I am sure you would like to read are: -. where \beta(s) is the Dirichlet beta Function. \begin{align} all even negative integers. Zeta. Is it right to replace Hamiltonian with Lagrangian in the Schrödinger equation? Your question: " Is there a high level understanding for this disparity between even and odd integers?" Where do the many many proofs powerful enough to evaluate \zeta(2n) stumble when it comes to evaluating \zeta(2n+1)? Was AGP only ever used for graphics cards. However, the latter are null (the so called simple "trivial zeroes"), and the mystery actually lies in the first terms of the Taylor expansions . &=\Res_{z=0}\frac{\pi\sec(\pi z)}{z^{2k+1}}+2\sum_{n=1}^\infty\frac{(-1)^n}{\left(\small{n-\frac12}\right)^{2k+1}}\\ button to find the value of the Riemann zeta fucntion at the specified point.. Computing \zeta(6)=\sum\limits_{k=1}^\infty \frac1{k^6} with Fourier series. Sci. is much deeper that you could think. There has been some progress in the form of Apery's theorem and other results such as "infinitely many of \zeta(2n+1) are irrational" or "at least one of \zeta(5),\zeta(7),\zeta(9) or \zeta(11) is irrational". Do I need HDMI-to-VGA or VGA-to-HDMI adapter? Written in this way, the zeta function at even integers reveals its alter ego as an Eisenstein series in one dimension. But for k odd this is equal to zero, since terms cancel with their negatives! The paper is not viable to read. \end{align}, Riemann zeta function at odd positive integers, Bernoulli Numbers and The Riemann Zeta function, A note on Value of the Riemann Zeta function at Odd Positive Integers, Creating new Help Center documents for Review queues: Project overview, Feature Preview: New Review Suspensions Mod UX, Different methods to compute $\sum\limits_{k=1}^\infty \frac{1}{k^2}$ (Basel problem).

This site uses Akismet to reduce spam. What situation would prompt the world to dump the use of Atomic and Nuclear Explosives entirely? &=\Res_{z=0}\frac{\pi\tan(\pi z)}{z^{2k}}+2\sum_{n=1}^\infty\frac{-1}{\left(\small{n-\frac12}\right)^{2k}}\\

This agrees with the fact that the residue of an even function at $z=0$ is $0$. Bernoulli Numbers and The Riemann Zeta function by B.Sury. Why do mathematicians care so much about zeta functions? Is it best to attack the flat before a hill? Is it possible Alpha Zero will eventually solve chess?

What circumstances could lead to city layout based on hexagons? Here is the link: Thanks for the wonderful explanation. This is good for summing odd functions.
&=\Res_{z=0}\frac{\pi\sec(\pi z)}{z^{2k+1}}+4^{k+1}\sum_{n=1}^\infty\frac{(-1)^n}{(2n-1)^{2k+1}}\\ s = 3, 5, 7, 9,...etc.) Why did the F of "sneeze" and "snore" change to an S in English history? \end{align} @Dinesh: nope. Need help finding intersection of a hyperbola and a circle. \sum_{n=0}^\infty\frac{(-1)^{nk}}{(2n+1)^k} (revision of 2012)  In this post, we evaluate even values of the Riemann zeta function. Or is the belief that such a closed form summation is unlikely?