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Let $k$ be the rational function field over the finite field of $q$ elementsand $\bar{k}$ its fixed algebraic closure. In this paper, we study algebraicrelations over $\bar{k}$ among the fundamental period $\widetilde{\pi}$ of theCarlitz module and the positive characteristic multizeta values $\zeta(n)$ and$\zeta(n,n)$ for an "odd" integer $n$, where we say that $n$ is "odd" if $q-1$does not divide $n$. We prove that either they are algebraically independentover $\bar{k}$ or satisfy some simple relation over $k$. We also prove that if$2n$ is "odd" then they are algebraically independent over $\bar{k}$.

Algebraic independence of the Carlitz period and the positive characteristic multizeta values at n and (n,n)