Expansion, divisibility and parity

Abstract

Let P ⊂ [H₀,H] be a set of primes, where log H₀ ≥(log H)^(2/3+ϵ). Let L = Σ_(pϵP)1/p). Let N be such that log H ≤ (log N)^(1/2- ϵ). We show there exists a subset X⊂(N, 2N] of density close to 1 such that all the eigenvalues of the linear operator (A_|Xf)(n) = Σ/pϵP:p|n/n,n±pϵX f(n±p) – Σ/pϵP/n±pϵX f(n±p)/p are O(√L). This bound is optimal up to a constant factor. In other words, we prove that a graph describing divisibility by primes is a strong local expander almost everywhere, and indeed within a constant factor of being "locally Ramanujan" (a.e.). Specializing to f(n) = λ(n) with λ(n) the Liouville function, and using an estimate by Matomaki, Radziwill and Tao on the average of λ(n) in short intervals, we derive that 1/log x Σ/(n≤xλ(n)λ(n+1)/n = 0(1/√log log x), improving on a result of Tao's. We also prove that Σ_(N

Files

Additional details