无平方支持函数的显式平均值：卷积方法的边缘
Explicit averages of square-free supported functions: to the edge of the convolution method
摘要：
我们给出了卷积方法的一般说明，以便可以提供明确的渐近估计的平方自由支持算术函数的所有平均值，有一个足够经常的顺序上的素数，并观察这种方法的性质如何给误差项估计的顺序$X^{-δ}$，其中$δ$属于一个开放的真正的积极的设置$I$。为了有一个更好的误差估计，一个自然的问题是我们是否可以实现一个临界阶的误差项$X^{-δ_0}$，其中$δ_0$，临界指数，是$I$的右端点。我们积极回答这个问题，提出了一种新的方法，提高了定性几乎所有的情况下的卷积方法在一定的正则性条件下，现在，性能良好的平方自由支持算术函数的平均值的渐近估计，可以给出其临界指数和一个合理的显式误差常数。我们通过分析与Ramaré-Akhilesh（2017）的工作相关的特定平均值来说明这种新方法，这在施加非平凡互质条件时会导致显着的改进。
英文摘要：
We give a general statement of the convolution method so that one can provide explicit asymptotic estimations for all averages of square-free supported arithmetic functions that have a sufficiently regular order on the prime numbers and observe how the nature of this method gives error term estimations of order $X^{-δ}$, where $δ$ belongs to an open real positive set $I$. In order to have a better error estimation, a natural question is whether or not we can achieve an error term of critical order $X^{-δ_0}$, where $δ_0$, the critical exponent, is the right hand endpoint of $I$. We reply positively to that question by presenting a new method that improves qualitatively almost all instances of the convolution method under some regularity conditions; now, the asymptotic estimation of averages of well-behaved square-free supported arithmetic functions can be given with its critical exponent and a reasonable explicit error constant. We illustrate this new method by analyzing a particular average related to the work of Ramaré--Akhilesh (2017), which leads to notable improvements when imposing non-trivial coprimality conditions.
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