Some basic results in elementary number theory in function fields

In this paper, we introduce a Carlitz module analogue of the classical Euler totient function, and prove a function field analogue of Euler's theorem by using the Carlitz action and the Carlitz module analogue of the Euler totient function. We propose a function field analogue of Carmichael's totient function conjecture. In contrast to the classical case, we answer the function field analogue of Carmichael's conjecture in the negative. We also propose a function field analogue of Sierpiński's conjecture, and discuss some special cases of this analogue.