On The Mean Estimate Of Chowla Conjecture

Let ? denote the Liouville function. From the PNT, we can know that A well known conjecture of Chowla asserts that for any distinct natural numbers h1,…,hk,one has as X??. This conjecture remains open for any h1,…,hk with k?2.In this paper, using the similar method of [1] —— a certain dense set S of natural numbers that have a "typical" prime factorization in a certain specific sense,we establish an averaged result of this conjecture, namely as X??. whenever H = H(X) ? X goes to infinity as X ? ?, ? <1/2500, and k is fixed, which improved improved the work of Matomaki, Radziwill, Tao [7].Related to this, we also give the exponential sum estimate with the ? < 1/625 which is an improvement over the result by Matomaki, Radziwill, Tao[7].