The Möbius function is strongly orthogonal to nilsequences

Abstract

We show that the Möbius function µ(n) is strongly asymptotically orthogonal to any polynomial nilsequence (F (g(n)Γ)) n∈N . Here, G is a simply-connected nilpotent Lie group with a discrete and cocompact subgroup Γ (so G/Γ is a nilmanifold ), g : Z → G is a polynomial sequence, andIn particular, this implies the Möbius and Nilsequence conjecture MN(s) from our earlier paper for every positive integer s. This is one of two major ingredients in our programme to establish a large number of cases of the generalised Hardy-Littlewood conjecture, which predicts how often a collection ψ1, . . . , ψt : Z d → Z of linear forms all take prime values. The proof is a relatively quick application of the results in our recent companion paper. We give some applications of our main theorem. We show, for example, that the Möbius function is uncorrelated with any bracket polynomial such as n √ 3 n √ 2 . We also obtain a result about the distribution of nilsequences (a n xΓ) n∈N as n ranges only over the primes.

An effective Ratner equidistribution result for SL(2,R)⋉R2

“…[5, §1.5.2] for a discussion; compare also p. 3 below). Recently, however, some new important cases have been established: Green and Tao [15] have proved effective equidistribution of polynomial orbits on nilmanifolds; this is an important input in their work on linear equations in primes [14], [16]. Moreover, Einsiedler, Margulis and Venkatesh [5] have proved effective equidistribution for large closed orbits of semisimple groups on homogeneous spaces; see also Mohammadi [34] for a more explicit result in the special case of closed SO(2, 1)-orbits in SL(3, Z)\ SL (3, R).…”

The Autocorrelation of the Mobius Function and Chowla's Conjecture for the Rational Function Field

“…This sum measures the correlation between µ(n) and the constant function. Recent studies have explored the correlation between µ(n) and other sequences, see [5,2,1]. Sarnak [8] showed that µ(n) does not correlate with any "deterministic" (i. e., zero entropy) sequence, assuming an old conjecture of Chowla [3] on the auto-correlation of the Möbius function, which asserts that given an r-tuple of distinct integers α 1 , .…”

0-1 sequences of the Thue-Morse type and Sarnak’s conjecture

“…In numerous cases, this conjecture has recently been proved to hold: [1], [3], [4], [9], [16], [19], [21] (see also [25]). …”