大篩其稀疏集合之元素為模數及其在平方模數之應用的新進展

Abstract

模數為平方數時的大篩不等式有以下形式： \[ \sum_{q=1}^Q\sum_{\substack{a~\mathrm{mod}~q^2\\\mathrm{gcd}(a,q)=1}} \left|\sum_{n= M+1}^{M+N} a_ne\left( \frac{a}{q^2}n\right) \right|^2\ll\Delta \sum_{n=M+1}^{M+ N} \left| a_n\right|^2 \]。由古典大篩不等式，我們可得兩個自然的$\Delta$，即$\Delta=Q^4+N$和$\Delta=Q(Q^2+N)$。趙良軼 \cite{Zhao 2004} 給出一個$\Delta$，即$\Delta=Q^3+(N\sqrt{Q }+\sqrt{N}Q^2)N^\varepsilon$，它在$N^{2/7+\varepsilon}\ll Q\ll N^{1/2-\varepsilon}$比以上兩個自然的$\Delta$準確。

延用D. Wolke \cite{Wolke 1971/2}的某些方法，對於某個稀疏集合$\mathcal{S}$其元素構成等差數列，S. Baier得到了一般形式的大篩不等式，即定理\ref {thm 2:Baier}。之後，令定理\ref{thm 2:Baier}的集合$\mathcal{S}$其元素為平方數，他推得定理\ref{thm 3:Baier}，這時$\Delta=(\log\log 10NQ)^3(Q^3+N+N^{1/2+\varepsilon}Q^2)$，它在$N^{1/4+\varepsilon}\ll Q\ll N^{1/ 3-\varepsilon}$比以上兩個自然的和趙良軼的$\Delta$準確。

The large sieve inequality for square moduli has the following form: \[ \sum_{q=1}^Q\sum_{\substack{a~\mathrm{mod}~q^2\\\mathrm{gcd}(a,q)=1}} \left|\sum_{n=M+1}^{M+N} a_ne\left( \frac{a}{q^2}n\right) \right|^2\ll\Delta \sum_{n=M+1}^{M+N} \left| a_n\right|^2. \] From the classical large sieve inequality, we can deduce two natural $\Delta$s, namely $\Delta=Q^4+N$ and $\Delta=Q(Q^2+N)$. L. Zhao \cite{Zhao 2004} gives a $\Delta$, namely $\Delta=Q^3+(N\sqrt{Q}+\sqrt{N}Q^2)N^\varepsilon$ in (\ref{Zhao's bound}), it is sharper than the former two $\Delta$s in the range $N^{2/7+\varepsilon}\ll Q\ll N^{1/2-\varepsilon}$.

Extending a method of D. Wolke \cite{Wolke 1971/2}, S. Baier \cite{Baier 2006} establishes a general large sieve inequality (see Theorem \ref{thm 2:Baier} below), for the case when $\mathcal{S}$ is a sparse set of moduli which is in a certain sense well-distributed in arithmetic progressions. As an application, he then employs Theorem \ref{thm 2:Baier} with $\mathcal{S}$ consists of squares. In this case, he obtains Theorem \ref{thm 3:Baier} with a $\Delta=(\log\log10NQ)^3(Q^3+N+N^{1/2+\varepsilon}Q^2)$, it is sharper than the two natural $\Delta$s and Zhao's bound (\ref{Zhao's bound}) within the range $N^{1/4+\varepsilon}\ll Q\ll N^{1/3-\varepsilon}$.