Vol. 9, No. 9, 2015

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An averaged form of Chowla's conjecture

Kaisa Matomäki, Maksym Radziwiłł and Terence Tao

Vol. 9 (2015), No. 9, 2167–2196

Let λ denote the Liouville function. A well-known conjecture of Chowla asserts that, for any distinct natural numbers h1,,hk, one has

1nXλ(n + h1)λ(n + hk) = o(X)

as X . This conjecture remains unproven for any h1,,hk with k 2. Using the recent results of Matomäki and Radziwiłł on mean values of multiplicative functions in short intervals, combined with an argument of Kátai and Bourgain, Sarnak, and Ziegler, we establish an averaged version of this conjecture, namely

h1,,hkH| 1nXλ(n + h1)λ(n + hk)| = o(HkX)

as X , whenever H = H(X) X goes to infinity as X and k is fixed. Related to this, we give the exponential sum estimate

0X| xnx+Hλ(n)e(αn)|dx = o(HX)

as X uniformly for all α , with H as before. Our arguments in fact give quantitative bounds on the decay rate (roughly on the order of loglogHlogH) and extend to more general bounded multiplicative functions than the Liouville function, yielding an averaged form of a (corrected) conjecture of Elliott.

multiplicative functions, Hardy–Littlewood circle method, Chowla conjecture
Mathematical Subject Classification 2010
Primary: 11P32
Received: 17 April 2015
Revised: 28 August 2015
Accepted: 6 October 2015
Published: 4 November 2015
Kaisa Matomäki
Department of Mathematics and Statistics
University of Turku
FI-20014 Turku
Maksym Radziwiłł
Department of Mathematics
Rutgers University
Hill Center for the Mathematical Sciences
110 Frelinghuysen Rd.
Piscataway, NJ 08854-8019
United States
Terence Tao
Department of Mathematics
University of California Los Angeles
405 Hilgard Avenue
Los Angeles, CA 90095-1555
United States