I am a 19 yr old student new to all these ideas. I made the transformation $X(z)=\sum_{n=1}^\infty z^n/n^2$. Therefore $X(1)=\pi^2/6$ as we all know (it is $\zeta(2)$). To calculate $X(1)$, I integrated
$$\frac{Y(z)}{z}=\sum_{n=1}^\infty \frac{z^{n-1}}{n} $$
between 0 to 1;

We all know $Y(z)=-\log(1-z)$; so doing integral of $Y(z)/z$ from 0 to 1, I get

$$X(1)=\int_0^1\ln\frac{z}{z-1}dz$$
that is $\pi^2/6$; Now integrating $X(z)/z$ between 0 to 1, I get $\sum_{n=1}^\infty 1/n^3$;

therefore performing that integral I get $\sum1/n^3$ as 
$$\int_0^1 \frac{(\log x)^2 dx}{1-x};$$
which Iam unable to do.

So can anyone give an idea of how to find 
$$\sum_{n=1}^\infty\frac1{n^3}=\zeta(3)=\int_0^1 \frac{(\log x)^2 dx}{1-x};$$
given
$$\zeta(2)=\int_0^1 \frac{\log x\ dx}{1-x}=\frac{\pi^2}{6}.$$
Please take time to read all these things and help me out by giving some suggestion.

Dear Vamsi,

Unlike the special values $\zeta(2n)$ (for $n \geq 1$), which are known to be simple algebraic expressions in $\pi$ (in fact just rational multiples of $\pi^{2n}$), it is conjectured (but not known) that the values $\zeta(2n+1)$ are genuinely new irrationalities (and that in fact
each is genuinely different from the other); more precisely, they are conjectured to be algebraically independent of one another, and of $\pi$.  There is no prior, classical, name
for these numbers, and in particular you should not expect to be able to evaluate your integral in terms of any numbers whose names you already know.

There is a theoretical basis for this conjecture: the kind of integrals that you are computing are callled "period integrals" (if you search, you will find a few other MO questions about periods, in this sense), and a general philosophy is that period integrals should have no more relations between them than those that are implied by elementary manipulations of integrals
(of the type that you made to compute $\zeta(2)$; and here I don't mean elementary in a disparaging sense, just in the sense of standard rules for computing integrals).    In fact, period integrals are manifestations of underlying geometry (which I won't get into here; all I will say is that the geometry relevant to zeta values is the geometry of "mixed Tate motives").  One can show that the geometric objects underlying the odd zeta values are independent of one another, in a suitable sense, and of the geometric object underlying $\pi$ (which is basically the circle); what is missing is a proof that the period integrals faithfully reflect the underlying geometry (so that independence in geometry implies independence of period integrals).  This is one of the big conjectures in contemporary arithmetic geometry and number theory, and so your question, which is a very nice one, is touching on some very fundamental (and difficult) mathematical issues.

Good luck as you continue your studies!

Best wishes,

Matthew Emerton  
