An averaged form of Chowla's conjecture

作者：Matomaki Kaisa^*; Radziwill Maksym; Tao Terence

来源：Algebra and Number Theory, 2015, 9(9): 2167-2196.

DOI：10.2140/ant.2015.9.2167

• multiplicative functions
• Hardy-Littlewood circle method
• Chowla conjecture

Let lambda denote the Liouville function. A well-known conjecture of Chowla asserts that, for any distinct natural numbers h(1), ... , h(k), one has
Sigma(1 <= n <= X) lambda(n+h(1))... lambda(n+h(k)) = o(X)
as X -> infinity. This conjecture remains unproven for any h(1), ... , h(k) with k >= 2. Using the recent results of Matomaki and Radziwill on mean values of multiplicative functions in short intervals, combined with an argument of Katai and Bourgain, Sarnak, and Ziegler, we establish an averaged version of this conjecture, namely
Sigma(h1, ..., hk <= H) vertical bar Sigma(1 <= n <= X) lambda(n+h(1))... lambda(n+h(k)) = o((HX)-X-k)
as X -> infinity, whenever H = H (X) <= X goes to infinity as X -> infinity and k is fixed. Related to this, we give the exponential sum estimate
integral(X)(0) vertical bar Sigma(x <= n <= x+H) lambda(n)e (alpha n) vertical bar dx = o(HX)
as X -> infinity uniformly for all alpha is an element of R, with H as before. Our arguments in fact give quantitative bounds on the decay rate (roughly on the order of log log H / log H) and extend to more general bounded multiplicative functions than the Liouville function, yielding an averaged form of a (corrected) conjecture of Elliott.

• 出版日期2015