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Published August 25, 2021 | Submitted
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Expansion, divisibility and parity


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

Additional Information

H. A. Helfgott was supported by his European Research Council Consolidator grant (Grant ID: 648329; codename GRANT) and by his Humboldt professorship. M. Radziwill was supported by a Sloan Fellowship and NSF grant DMS-1902063. The authors also thank MSRI (Berkeley) and AIM (San Jose) as well as their home institutions for hosting them during visits. They are grateful to several colleagues who gave them helpful answers and references, mostly on MathOverflow: Yves Cornulier, Hailong Dao, R. van Dobben de Bruyn, Shmuel Friedland, Oleksiy Klurman, Dimitris Koukoulopoulos, Achim Krause, Lek-Heng Lim, Michael Magee, Brendan McKay, Anton Mellit, Ryan O'Donnell, Fedor Petrov, Federico Poloni, Geoff Robinson, Will Sawin, Ilya Shkredov, Lior Silberman, Gerald Tenenbaum, Adrian Ubis, Andre Uschmajew and Gjerji Zaimi, and pseudonymous users BS., MTyson, user174768, user174996, vidyarthi and 2734364041, among others. They would also like to thank Kaisa Matomaki for early discussions and later helpful remarks. H. A. Helfgott would also like to express his deep appreciation to those graduate students and postdocs at Göttingen who, during the COVID-19 pandemic, attended two semester-long virtual lecture courses he gave on the proof as it was still taking shape.

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