# Prime and Möbius correlations for very short intervals in $F_p[x]$

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Speaker:
Pär Kurlberg
Affiliation:
KTH Stockholm
Date:
Tue, 2018-09-04 15:05 - 15:55
Location:
MPIM Lecture Hall

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