Special relations between multizeta values and parity results

We study relations between multizeta values for function fields introduced by D. Thakur. The F_p-span of Thakur's multizeta values is an algebra (Thakur. Shuffle relations for function field multizeta values). In particular, the product \zeta(a)\zeta(b) is a linear combination of multizeta values. In this paper, several of the conjectures formulated by the author and by D. Thakur for small values or for special families of a about how to write \zeta(a)\zeta(b) as an F_p-linear combination of multizeta values, are proved. Also, the parity conjecture formulated by Thakur is proved.