【作者】 张伟；

【导师】 唐恒才；

【作者基本信息】 河南大学 ， 基础数学， 2017， 硕士

【摘要】 设λ表示Liouville函数.当X →∞时,由素数定理可知对任意互不相同的自然数h1,…,hk,k≥2时,问题到现在依然没有得到解决,这就是著名Chowla猜想.本文应用文献[6]中关于筛法的关键技术,优化了 Matomaki, Radziwill , T[7]关于Chowla猜想均值估计的上界估计,即当X →∞,且H=H(X)≤X时,我们有其中k是一个固定的常数,θ<1/2500.在Matomaki, Radziwill, Tao [7]的论文中θ=1/3000.我们同时也优化了相应的指数和估计,其中η <1/625.在Matomaki, Radziwill, Tao [7]的工作中η=1/700.更多还原

【Abstract】 Let λ 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 → ∞, θ <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 η < 1/625 which is an improvement over the result by Matomaki, Radziwill, Tao[7].更多还原