We investigate function field analogs of the distribution of primes,

and prime $k$-tuples, in "very short intervals'' of the form $I(f) :=

\{ f(x) + a : a \in F_p \}$ for $f(x) \in F_p[x]$ and $p$ prime, as

well as cancellation in sums of function field analogs of the Möbius

$\mu$ function and its correlations (similar to sums appearing in

Chowla's conjecture).

For generic $f$, i.e., for $f$ a "Morse polynomial", we show that

error terms are roughly of size $O(\sqrt{p})$ (with typical main terms

of order $p$). We also give examples of $f$ for which there is no

cancellation at all, and intervals where the heuristic ``primes are

independent'' fails very badly.

Time permitting we will discuss the curious fact that (square root)

cancellation in Möbius sums is **equivalent** to (square root)

cancellation in Chowla type sums.

© MPI f. Mathematik, Bonn | Impressum & Datenschutz |