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Expansion, divisibility and parity

Helfgott, Harald Andrés and Radziwiłł, Maksym (2021) Expansion, divisibility and parity. . (Unpublished)

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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<n≤2Nλ(n) λ(n+1)=o(N) at almost all scales with a similar error term, improving on a result by Tao and Teravainen. (Tao and Tao-Teravainen followed a different approach, based on entropy, not expansion; significantly, we can take a much larger value of H, and thus consider many more primes.) We can also prove sharper results with ease. Thus, for instance, we can show that, 1/log x Σ/n≤x|Ω(n)–Ω(n+1)|≤s(x) λ(n) λ(n+1)/n = o(s(x)/√log log x) for any s(x) tending to ∞ as → ∞, where Ω(n) is the number of prime divisors of n, considered with multiplicity. We also show that, for example, for S_(N,k) the set of N < n ≤ 2N such that Ω(n) = k, and any fixed value of k with k = log log N + O(√log log n), the average of λ(n + 1) over S_(N,k) is o(1) at almost all scales.

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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.
Funding AgencyGrant Number
European Research Council (ERC)648329
Georg August University of GöttingenUNSPECIFIED
Alfred P. Sloan FoundationUNSPECIFIED
Record Number:CaltechAUTHORS:20210825-184543983
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:110534
Deposited By: George Porter
Deposited On:25 Aug 2021 23:01
Last Modified:25 Aug 2021 23:02

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