Algebraic independence of the Carlitz period and the positive characteristic multizeta values at 𝑛 and (𝑛,𝑛)
Yoshinori Mishiba · Proceedings of the American Mathematical Society · 2015
Let k k be the rational function field over the finite field of q q elements and k ¯ \overline {k} its fixed algebraic closure. In this paper, we study algebraic relations over k ¯ \overline {k} among the fundamental period π ~ \widetilde {\pi } of the Carlitz module and the positive characteristic multizeta values ζ ( n ) \zeta (n) and ζ ( n , n ) \zeta (n,n) for an “odd” integer n n , where we say that n n is “odd” if q − 1 q-1 does not divide n n . We prove that these three elements are either algebraically independent over k ¯ \overline {k} or satisfy some simple relation over k k . We also prove that if 2 n 2n is “odd”, then these three elements are algebraically independent over k ¯