Riemann zeta function at odd positive integers

The zeta function is defined as a sum over the positive integers, but as far as actually evaluating it, it turns out to be more natural to think of it as a sum over all nonzero integers; thus we should really be thinking about $\sum_{n \neq 0} \frac{1}{n^k}$. For $k$ even this is just $2 \zeta(k)$ and there are various ways to evaluate this more symmetric sum, e.g. by writing down a meromorphic function with the right poles, or a Fourier series with the right coefficients, etc. But for $k$ odd this is equal to zero, since terms cancel with their negatives! Written in this way, the zeta function at even integers reveals its alter ego as an Eisenstein series in one dimension.

This cancellation phenomenon occurs in Euler's classic "proof," since the infinite product for $\sin z$ that he uses has zeroes at all integer multiples of $\pi$, not just the positive ones. It also occurs in the general proof that proceeds by considering the generating function $\frac{z}{e^z - 1}$ for the Bernoulli numbers. As you might know, the closed form of $\zeta(2k)$ involves Bernoulli numbers, and again $\frac{z}{e^z - 1}$ has poles at $2 \pi i n$ for all nonzero integers $n$, not just the positive ones. I describe how this works in slightly more detail here.

Another way to think about the difference between the even and odd cases is that one can think of the even cases as $L^2$ norms of appropriate Fourier series; this is precisely how a standard proof of the evaluation of $\zeta(2)$ works. But for the odd cases we don't get an $L^2$ norm; instead we get a mysterious inner product.

One expects that the numbers $\zeta(2n+1)$ are algebraically independent of one another, and of $\pi$, and so one should think of them as ``new'' numbers; you can't expect any closed form expression in terms of powers of $\pi$, say. Unfortunately, this conjecture seems very much out of reach at the moment.

For an explanation of why people believe this conjecture, one can see for example this answer to a related mathoverflow question.

Have you looked at this ICM paper before. Ramanujan has found out some wierd formula. The paper is not viable to read. But I guess, this will give some idea about the progress made regarding this problem.

Added: Try emailing Prof. Bruce Berndt. Since this is related to Ramanujan, I am sure he might be knowing something along these lines. Another good source of information which I am sure you would like to read are: -

• Bernoulli Numbers and The Riemann Zeta function by B.Sury.

• A note on Value of the Riemann Zeta function at Odd Positive Integers by Andrzej Dabrowski.