THE AUTOCORRELATION OF THE MOBIUS FUNCTION AND CHOWLA'S CONJECTURE FOR THE RATIONAL FUNCTION FIELD

Abstract

We prove a function field version of Chowla's conjecture on the autocorrelation of the Möbius function in the limit of a large finite field.Note that the number of nonzero summands here, that is the number of n ≤ N for which n + α 1 , . . .n + α r are all square-free, is asymptotically c(α)N , where c(α) > 0 if the numbers α 1 , . . ., α r do not contain a complete system of residues modulo p 2 for every prime p [6], so that (1.1) is about non-trivial cancellation in the sum.Chowla's conjecture (1.1) seems intractable at this time, the only known case being r = 1 where it is equivalent with the Prime Number Theorem.Our goal in this note is to prove a function field version of Chowla's conjecture.Let F q be a finite field of q elements, and F q [x] the polynomial ring over F q .The Möbius function of a nonzero polynomial F ∈ F q [x] is defined to be µ(F ) = (-1) r if F = cP 1 . . .P r with 0 = c ∈ F q and P 1 , . . ., P r are distinct monic irreducible polynomials, and µ(F ) = 0 otherwise.

Locations

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